First off - I am not professionally involved with science, nor medicine. But I heard the same statement, that "flu vaccine make us more vulnerable to COVID-19" and was also curious. Here are the results of my research.
Also - there is a conclusion in the end if you don't want to read the whole post.
Research papers that are saying flu vaccine make you less vulnerable to COVID-19
There are research papers that suggest lower mortality from COVID-19 among people who were vaccinated against influenza, and lower chances for serious cases. For example in Brazil:
We analyzed data from 92,664 clinically and molecularly confirmed Covid-19 cases in Brazil to understand the potential associations between influenza vaccination and Covid-19 outcomes. Controlling for health facility of treatment, comorbidities as well as an extensive range of sociodemographic factors, we show that patients who received a recent influenza vaccine experienced on average 8% lower odds of needing intensive care treatment (95% CIs [0.86, 0.99]), 18% lower odds of requiring invasive respiratory support (0.74, 0.88) and 17% lower odds of death (0.75, 0.89). Large scale promotion of influenza vaccines seems advisable
Inactivated trivalent influenza vaccine is associated with lower mortality among Covid-19 patients in Brazil
or among older people in the USA:
we explore a possible area-level association between influenza vaccination coverage in people aged 65 years and older and the number of deaths from COVID-19. [...] We adjusted for a wide array of potential confounding variables using both county-level generalized propensity scores for influenza vaccination rates, as well as direct adjustment. Our results suggest that influenza vaccination coverage in the elderly population is negatively associated with mortality from COVID-19
Influenza Vaccination and COVID19 Mortality in the USA
The flu vaccine will not stop you from getting COVID-19, but some sources as I presented suggest that it may reduce the chances of serious COVID-19 cases. Why? It may be because vaccine stimulates non-specific immune mechanisms on mucous membranes (Sorry, but the source for this statement comes from a polish YouTuber. He bases his speech on research articles that I linked above and his education in medicine: Polish YouTuber video ).
In fact, research from the USA is mentioning the "DoD personnel research" that Total Health Matters article that you found is based on:
On one hand, seasonal influenza infection is associated with the development of several respiratory complications, especially in the elderly. On the other hand, a recent report points to increased odds of non-SARS coronaviruses infections among army personnel who received influenza vaccination. This observation was attributed by the authors to the possibility that vaccinated individuals may lack the non-specific immunity acquired by natural influenza infection, which would protect against infection by other viruses.
[...] In our analyses, we controlled for a wide array of potential confounding variables, including population density, social and economic variables, education, chronic medical conditions [...]. Our results suggest a reduction in COVID-19 mortality associated with higher influenza vaccination rates in the elderly population. Specifically, we found that overall, a 10% increase in vaccination coverage was associated on average with a statistically significant 28% decrease in the COVID-19 death rate. [...] additional confirmatory studies at the individual level are urgently needed.
So it seems like they wanted to check if flu vaccines can influence how older people are susceptible to COVID-19 and it turns out that in the older population in the USA it is associated with a significantly lower death rate, there is no effect suggested by Total Health Matters. But more research is needed.
Remember, that study on Department of Defense personnel that suggested flu vaccine might increase the risk of coronaviruses infections was carried out in the 2017-2018 flu season and so it was about other "Non-SARS" coronaviruses. Not SARS-Cov-2, and it was not even about SARS-Cov-1, just "regular" coronaviruses that are causing common cold.
Also, it is worth mentioning in this context, that not getting the flu shot increase the risk of being ill with regular influenza, and your body is more susceptible to other infections after it went through influenza - infections from various bacteria and viruses, like SARS-Cov-2. Yale Scientific - Influenza Renders Immune System Vulnerable
Keep in mind officials might be recommending flu vaccines not only because your body is weaker after the flu, and more susceptible to Sars-Cov-2 (and to many other viruses and bacteria that are causing "common" infections of the upper respiratory system) but also because it will reduce the burden on the healthcare system in the flu season, so it may focus on helping COVID-19 patients. (this is only my personal opinion, with no proof that this is one of the reasons behind officials recommendations).
Research papers that are saying flu vaccine might make you more vulnerable
On the other hand, Department of Defense research is not the only research paper that suggests flu vaccines might make you more susceptible to Sars-Cov-2. There is research that looked at statistical data from OECD countries and concluded that the influenza vaccine might increase the risk of getting SARS-Cov-2. It says that the reason might be "antibody-dependent enhancement" (ADE), a phenomenon which might make you more susceptible to other viruses after your body created antibodies for the flu (or other things that you vaccinated yourself for, or just git sick with). But they have no proof that ADE is taking place, they just did statistical analysis and are guessing why some countries have higher infection rates. Association between Influenza Vaccination Rates and SARS-CoV-2 Outbreak Infection Rates in OECD Countries.
Antibody-dependent enhancement is a real phenomenon and has been observed with dengue virus, Zika virus, Ebola virus and it might be an impediment in creating an effective vaccines for some illnesses (unfortunately it might also make it harder to create an effective and safe vaccine for COVID-19. News Feature: Avoiding pitfalls in the pursuit of a COVID-19 vaccine, also: Implications of antibody-dependent enhancement of infection for SARS-CoV-2 countermeasures)
But you need to keep in mind, that if antibody-dependent enhancement is a problem with flu and SARS-Cov-2 (and there is no proof that it is), you might be the "victim" of it not only after getting the flu shot but also after getting the flu itself, since your body produces flu antibodies after getting sick with flu. In the second option (not vaccinating, but getting sick with flu) in addition to antibody-dependent enhancement of SARS-Cov-2 your body will be additionally weakened by the flu that you just fought, and it's not good for you, of course. (see "Influenza Renders Immune System Vulnerable" link above).
And this research is pointing only to correlation, while there are many more reasons (more correlates, other than flu vaccinations) why some OECD countries might have higher mortality or higher infection rate. One of them is the fact, that some countries did not have compulsory vaccination against tuberculosis, and there are research papers that suggest tuberculosis vaccination reduces COVID-19 mortality.
Correlation between universal BCG vaccination policy and reduced mortality for COVID-19.
Most big studies draw a conclusion that the flu vaccine will make you less likely to get serious COVID-19 case after having contact with SARS-Cov-2. It is important that the same conclusion applies to older people, who are in the risk group for SARS-Cov-2, that's why they should vaccine against flu, since they are also in the risk group for serious flu complications (and flu complications further decrease your chances of getting alive out of COVID-19). Vaccinating will reduce your chances of getting the flu, and thus reduce your chances of weakening your immune system for many days, by the flu.
To my knowledge, much fewer studies suggest that flu vaccine might make you more susceptible for COVID-19, and they lack hard evidence. One research that is widely quoted by so called "alternative medicine" sources was about "regular" coronaviruses, not SARS-Cov-2 (research from 2017-2018 on army/DoD personnel). Other research is based on correlation while not paying attention to other factors than flu shot, that might be the reason of that correlation, like lack of BCG vaccinations in countries with higher mortality of COVID-19.
These research papers suggests that "antibody-dependent enhancement" (ADE) might be the reason of alleged higher chance of COVID-19 after getting flu shot, but there is no proof that it is the case with flu vaccine and SARS-Cov-2. And if ADE would be possible with flu shot and SARS-Cov-2, you might also have "antibody-dependent enhancement" of SARS-Cov-2 after not vaccinating and getting sick with the flu, but then the flu might make it even worse because your body will be weaker after fighting with it.
That's why my personal opinion is - right now problem with ADE is more like a possibility, not something confirmed, based on proofs or even worthy to be afraid of by average person who wants to take a flu shot. Risks of not vaccinating are bigger than this hypothetical risk, especially for people who are in the "risk zone" (older people, people that often have contact with influenza and have weak immune system) and this is what should scary people. Especially that we have stronger evidence that flu shot is not making you more susceptible to COVID-19 and might make you even less susceptible.