As discussed by Brian the claim is somewhat vague in that it states the theoretical possibility of something and then asserts the existence of something else.
But if take the claim to be existence of such viruses with on-purpose [GoF] enhanced transmissibility, it is true, although the claim (interpreted that way) exaggerates the deadliness of the actual feats (at least of those published). From a 2015 workshop/review of GoF
Dr. Yoshihiro Kawaoka, from the University of Wisconsin-Madison, classified types of GoF research depending on the outcome of the experiments. The first category, which he called “gain of function research of concern,” includes the generation of viruses with properties that do not exist in nature. The now famous example he gave is the production of H5N1 influenza A viruses that are airborne-transmissible among ferrets, compared to the non-airborne transmissible wild type. The second category deals with the generation of viruses that may be more pathogenic and/or transmissible than the wild type viruses but are still comparable to or less problematic than those existing in nature. Kawaoka argued that the majority of strains studied have low pathogenicity, but mutations found in natural isolates will improve their replication in mammalian cells. Finally, the third category, which is somewhere in between the two first categories, includes the generation of highly pathogenic and/or transmissible viruses in animal models that nevertheless do not appear to be a major public health concern. An example is the high-growth A/PR/8/34 influenza strain found to have increased pathogenicity in mice but not in humans.
It's actually not totally clear which particular paper that highlighted sentence refers to. It was a hot topic of research with several papers published by different groups within a fairly short time frame. Some used a directed evolution approach while others did something closer to the "synthesized" (reassortment/chimeric) approach. A (2012) paper on enhanced ferret H5 influenza (using the directed evolution approach) is pretty cited. However as far as (observed) deadliness goes...
None of the recipient ferrets died after airborne infection with the mutant A/H5N1 viruses.
On the other hand, the virus might be more dangerous to humans; in fact this was the motivation for the study. (But as another nitpick, this virus was obtain by directed evolution rather than merely "synthesizing" two viruses together.)
Highly pathogenic avian influenza A/H5N1 virus can cause morbidity and mortality in humans but thus far has not acquired the ability to be transmitted by aerosol or respiratory droplet (“airborne transmission”) between humans. To address the concern that the virus could acquire this ability under natural conditions, we genetically modified A/H5N1 virus by site-directed mutagenesis and subsequent serial passage in ferrets. The genetically modified A/H5N1 virus acquired mutations during passage in ferrets, ultimately becoming airborne transmissible in ferrets.
The introduction of receptor-binding site mutations [...] in HA [hemagglutinin], acquired during ferret passage, did not result in increased cross-reactivity with human antisera [...], indicating that humans do not have antibodies against the HA of the airborne-transmissible A/H5N1 virus that was selected in our experiments.
More consistent with the "synthesizing" approach stated in the OP's question, a year later, another group did research how such H5N1 mutations (enhancing transmissibility) could occur by recombination with (human transmissible) H1N1 in pig-like animals; the actual research didn't wait for these re combinations to occur naturally, by tried out a whole bunch of them "in a dish":
Using reverse genetics, we systematically created 127 reassortant viruses between a duck isolate of H5N1, specifically retaining its hemagglutinin (HA) gene throughout, and a highly transmissible, human-infective H1N1 virus. We tested the virulence of the reassortants in mice as a correlate for virulence in humans and tested transmissibility in guinea pigs, which have both avian and mammalian types of airway receptor. Transmission studies showed that the H1N1 virus genes encoding acidic polymerase and nonstructural protein made the H5N1 virus transmissible by respiratory droplet between guinea pigs without killing them. Further experiments implicated other H1N1 genes in the enhancement of mammal-to-mammal transmission, including those that encode nucleoprotein, neuraminidase, and matrix, as well as mutations in H5 HA that improve affinity for humanlike airway receptors. Hence, avian H5N1 subtype viruses do have the potential to acquire mammalian transmissibility by reassortment in current agricultural scenarios.
Commentary in Nature on related work and implications:
Virologists have created H5N1 reassortants before. One study found that H5N1 did not produce transmissible hybrids when it reassorts with a flu strain called H3N24. But in 2011, Stacey Schultz-Cherry, a virologist at St. Jude Children's Research Hospital in Memphis, Tennessee, showed that pandemic H1N1 becomes more virulent if it carries the HA gene from H5N1. [...]
Chen’s team mixed and matched seven gene segments from H5N1 and H1N1 in every possible combination, to create 127 reassortant viruses, all with H5N1’s HA gene. Some of these hybrids could spread through the air between guinea pigs in adjacent cages, as long as they carried either or both of two genes from H1N1 called PA and NS. Two further genes from H1N1, NA and M, promoted airborne transmission to a lesser extent, and another, the NP gene, did so in combination with PA.
“It’s a very extensive paper,” says Schultz-Cherry. “It really shows that it’s more than just the HA. The other proteins are just as important and can drive transmission.” Chen says that health organisations should monitor wild viruses for the gene combinations that her team identified in the latest study. “If those kinds of reassortants are found, we’d need to pay high attention.”
[subsection: Knowledge gap]
It is unclear how the results apply to humans. Guinea pigs have bird-like receptor proteins in their upper airways in addition to mammalian ones, so reassortant viruses might bind in them more easily than they would in humans.
And scientists do not know whether the hybrid viruses are as deadly as the parent H5N1. The hybrids did not kill any of the guinea pigs they spread to, but Chen says that these rodents are not good models for pathogenicity in humans.
Although more comprehensive, this [Chinese Science study] was actually the 2nd study of this (reassortment) kind; it was "beaten" by a Japanese-US study published in Nature in 2012:
We identified a reassortant H5 HA/H1N1 virus-comprising H5 HA (from an H5N1 virus) with four mutations and the remaining seven gene segments from a 2009 pandemic H1N1 virus-that was capable of droplet transmission in a ferret model.
And by a 2011 US (Schultz-Cherry) study, which was mentioned in the Nature editorial:
A novel H1N1 influenza virus emerged in 2009 (pH1N1) to become the first influenza pandemic of the 21st century. This virus is now cocirculating with highly pathogenic H5N1 avian influenza viruses in many parts of the world, raising concerns that a reassortment event may lead to highly pathogenic influenza strains with the capacity to infect humans more readily and cause severe disease. To investigate the virulence of pH1N1-H5N1 reassortant viruses, we created pH1N1 (A/California/04/2009) viruses expressing individual genes from an avian H5N1 influenza strain (A/Hong Kong/483/1997). [...] these data suggest that reassortment between cocirculating human pH1N1 and avian H5N1 influenza strains will result in a virus with the potential for increased pathogenicity in mammals.
Regarding the less concerning results mentioned in that 2015 workshop/review para, it's worth noting that it's not always obvious what you'll get (in terms of pathogenicity) until you perform an experiment, e.g.
A conserved step of CoV replication is the translation and processing of replicase polyproteins containing 16 nonstructural protein domains (nsp's 1 to 16). The CoV nsp5 protease (3CLpro; Mpro) processes nsp's at 11 cleavage sites and is essential for virus replication. [...] However, the intra- and intermolecular determinants of nsp5 activity and their conservation across divergent CoVs are unknown, in part due to challenges in cultivating many human and zoonotic CoVs. To test for conservation of nsp5 structure-function determinants, we engineered chimeric betacoronavirus murine hepatitis virus (MHV) genomes encoding nsp5 proteases of human and bat alphacoronaviruses and betacoronaviruses. Exchange of nsp5 proteases from HCoV-HKU1 and HCoV-OC43, which share the same genogroup, genogroup 2a, with MHV, allowed for immediate viral recovery with efficient replication albeit with impaired fitness in direct competition with wild-type MHV.
So in this latter case/example the chimeric virus (a "mix" of common cold and MHV) turned out to spread less well than the wild type MHV. (Which is why papers like the latter with "negative results" don't make the OMG NEWS.) But I don't know if (and rather doubt that) that would have been easily predictable without bothering with the actual experiment.
In contrast, more recently and "doing the rounds" in the conspiracy circles during this Covid-19 outbreak is a line of research from Baric's lab (with collaboration on some papers with the Wuhan lab) on a chimeric CoVs that can [very likely] infect humans. (Obviously, this kind of research was only tried those chimeras on human cells in vitro and in transgenic mice expressing human receptors.) The researchers argue that doing these kinds of experiments is essential in order to be able to predict their probability of occurrence "in the wild"; from a (2016) review-ish paper of Baric et al.:
The emergence of severe acute respiratory syndrome coronavirus (SARS-CoV) and Middle East respiratory syndrome (MERS)-CoV highlights the continued risk of cross-species transmission leading to epidemic disease. This manuscript describes efforts to extend surveillance beyond sequence analysis, constructing chimeric and full-length zoonotic coronaviruses to evaluate emergence potential. Focusing on SARS-like virus sequences isolated from Chinese horseshoe bats, the results indicate a significant threat posed by WIV1-CoV. Both full-length and chimeric WIV1-CoV readily replicated efficiently in human airway cultures and in vivo, suggesting capability of direct transmission to humans. In addition, while monoclonal antibody treatments prove effective, the SARS-based vaccine approach failed to confer protection.
Again, knowing the outcome of the research before conducting it is quite doubtful, but if you put your tinfoil hat on, the fact that the chimeric virus turned out to be unaffected by the (experimental) SARS vaccine is OMG NEWS in the sense that "look they've created something more dangerous (in a sense) than SARS".
Note however that rather than focusing on enhancing some known human pathogen (the focus of the OP's claim), this relied on basically attempting zoonosis. It's not a given that such a resulting virus is better at replicating in humans than something that's already known. And in fact from this perspective the latter experiment is also "a flop":
However, pathogenesis studies in mice suggest that further adaptation may be required for epidemic disease. Compared with SARS equivalents, both full-length and chimeric WIV1 viruses had significant attenuation even with the presence of human ACE2 in the mouse model. Together, the data suggest that despite using ACE2 and robust replication in primary human airway epithelial cultures, WIV1-CoV likely maintains deficits that impact pathogenesis in mice; therefore, WIV1-mediated infection may have diminished epidemic potential in humans relative to SARS-CoV.
So actual experiments that do what the OP's claim says "take a virus which is quite deadly [...] and mix that with the part of virus that makes other viruses transmissible" haven't actually been done much (if at all) at least in published research. The (three at least in 2011-2013) studies on enhancing/recombining H5N1 (more deadly to us) with H1N1 (spreads more easily in us) seems to be the only one(s) technically fitting the bill that I found. (Actually, a bit more search finds a 2018 French paper on that too.) I would not be surprised however some other [more] deadly viruses like H7N9 have been similarly studied in a reassortment/chimeric context with the more transmissible human H1. However, this type of experiment for more distant viruses (as suggested in the OP's quote) doesn't seem to have been published; there surely be even more ethical concerns with conducting such experiments with viruses that are not co-occurring in nature, for which a reassortment research would be harder to justify.
On the bioweapon aspect of chimeric/reassortment viruses, I've only found a short section in a 2012 paper by an Israeli author, Dany Shoham. The most relevant/concrete bit was:
The yellow fever virus – a past standardized BW in the US Army, then
carried by infected Aedes mosquitoes as vectors – was thus genetically engineered in conjunction with IAV, resulting in chimeric virions with infectious capacity for different biological systems (Oliveira et al., 2002).
Further, the anthrax protective antigen, which is
one of the four proteins comprising the anthrax toxin,
has been produced by an infective chimeric influenzaanthrax hybrid, thereby bringing about neutralizing antibody response against the toxin in mice (William et al., 2010). Similarly, a chimeric influenza virus expressing an
epitope of outer membrane protein F of Pseudomonas
aeruginosa was created and afforded protection against
challenge with P. aeruginosa in a murine model (Staczek
et al., 1998).
Besides yellow fever, a variety of pathogenic viruses
have been engineered in conjunction with IAV. Functional
chimeras were thus obtained between HIV type 1 Gp120
and IAV hemagglutinin; (Copeland, 2005) highly pathogenic avian influenza virus and murine leukemia virus;
(Hatziioannou et al., 1998) hepatitis C virus E2 glycoprotein and IAV hemagglutinin; (Flint et al., 1999) and the
gC glycoprotein of herpes simplex virus with another AIV
HA (Lazarovits et al., 1996). All in all, the mastering of
those remarkably developing techniques strongly illustrates the versatile applicability of IAV, and might lead to
highly advanced military-oriented modifications.
As a point of caution, the same author has recently claimed that China has engaged in hidden bioweapon work in dual-use laboratories (and he included the one in Wuhan) in that list. China denies doing bioweapon research.
"Certain laboratories in the institute have probably been engaged, in terms of research and development, in Chinese (biological weapons), at least collaterally, yet not as a principal facility of the Chinese BW alignment," Shoham told The Washington Times.
Work on biological weapons is conducted as part of a dual civilian-military research and is "definitely covert," he said.
The info he gave about the yellow-fever IAV seems to check out (albeit from a pretty obscure Brazilian paper with only 3 citations in Google Scholar), but note that this was not a true genetic hybrid as the yellow fever virus was artificially coated with the IAV envelope, but would not have this feature once it reproduced on its own:
In order to obtain a better understanding of the functional mechanisms involved in the fusogenesis of enveloped viruses, the influenza A (X31) and the yellow fever (17DD) virus particles were used to construct a chimeric structure based on their distinct pH requirements for fusion, and the distinct malleability of their nucleocapsids. The malleable nucleocapsid of the influenza A virus particle is characterized by a pleomorphic configuration when observed by electron microscopy. A heat inactivated preparation of X31 virus was used as a lectin to interact with the sialic acid domains present in the 17DD virus envelope. The E spikes of 17DD virus were induced to promote fusion of both envelopes, creating a double genome enveloped structure, the chimeric yellow fever-influenza A virus particle. These chimeric viral particles, originally denominated 'partículas virais quiméricas' (PVQ), were characterized by their infectious capacity for different biological systems. Cell inoculation with PVQ resulted in viral products that showed similar characteristics to those obtained after 17DD virus infections.
On the other hand, Shoham's claim that "Functional chimeras were thus obtained between HIV type 1 Gp120 and IAV hemagglutinin; (Copeland, 2005)" is a bit more misleading, as the actual research/experiment only involved creating one chimeric protein on a vaccinia substrate.
On the harder-to-verify side, a USAF document states/paraphrases:
In late 1992, [...] Dr. Kanatjan
Alibekov became the third defector from the Russian BW program.
the Deputy Director (number-two man) of Biopreparat and an infectious
disease physician/epidemiologist, he was the highest ranking defector ever
from the program. (Dr. Alibekov anglicized his name and now goes by
Ken Alibek.) In 1999, Alibek published Biohazard, a first-hand detailed
account of his experiences. Alibek disclosed a virtual encyclopedia of
intimate details on Biopreparat from the top down [...]
In his book Biohazard, Alibek wrote about using plasmids to increase
virulence or antibiotic resistance in bacteria.
Pasechnik‘s and ―Temple Fortune‘s‖ prior statements. He also discussed
transfer of a gene for myelin toxin to Yersinia pestis (plague bacteria),
however this agent was reportedly not yet weaponized. He said that a new
Moscow-based company named Bioeffekt Ltd. had offered, by mail order,
three strains of tularemia produced by ―technology unknown outside
Russia‖ (i.e., genetically engineered strains).
Most astounding of all, Alibek revealed that genetic engineering
research was underway to create entirely new life forms.
The goal of hybrid ―chimera viruses was to insert genes from one virus into another
to create an even more lethal virus. Alibek stated that the Russians had
created the first chimera virus from inserting DNA from Venezuelan
equine encephalitis (VEE) virus into vaccinia virus (genetic structure
almost identical to the smallpox virus).
Chimeras, of VEE, Ebola, and
Marburg genes inserted into the actual smallpox virus, were in the research
phase when he left in 1991.
On the other hand, Wikipedia casts some doubt on that, also from US sources:
Retired Army major general and physician Philip K. Russell, while impressed by Alibek's knowledge of the former Soviet Union's production of anthrax, "began to think that Ken was more fanciful than precise in some of his recollections" where genetically engineered smallpox was concerned. Russell also remarked on "... the issue of putting Ebola genes into smallpox virus. That was viewed, at least in many of our minds, as somewhat fanciful. And probably not true."
Regarding the threat-level posed such technology (virus enhancement--by whatever means) if used "in the wrong hands", the US had fairly heated discussion in 2012 following the first H5 papers; eventually both those initially controversial papers were published without any parts being censored.
More recently (2018)
A new analysis of synthetic biologic threats commissioned by the US Department of Defense (DoD) found that three capabilities raise the greatest concerns: re-creating known pathogenic viruses, making existing bacteria more dangerous, and using microbes to produce harmful biochemical in humans. [...]
Note that enhancing viruses wasn't too far behind, but on a 2nd tier of concerns:
Making existing viruses more dangerous is on the second-highest tier, alongside making chemicals or biochemical by exploiting natural metabolic pathways. The lowest tier includes modifying the human genome using human gene drives.