In essence this is all based on the self-published (November 2021) paper by political scientist Beattie. To quote the main problem with it:
Beattie claimed to have analyzed the “causal impact” of COVID-19 vaccines on COVID-19 cases and deaths. To do this, he obtained data from Our World in Data, a public COVID-19 database, for the total cases and deaths in more than 140 countries for the past 12 to 16 months before vaccine administration began in those countries. He then used the data to create projections of what the numbers of cases and deaths at the current period would have been like without the vaccines. He then compared these numbers with the actual cases and deaths after COVID-19 vaccines came into use. This type of analysis, which uses aggregated data at the population level, is also known as an ecological study.
Because his analysis showed that countries had a higher number of cases and deaths after COVID-19 vaccines came into use, relative to the projected numbers based on the pre-vaccine period, he concluded that the vaccines were the cause of these increases.
But this conclusion is erroneous. Beattie’s analysis is fundamentally flawed because he failed to account for other variables that also directly affect cases and deaths (confounding factors), such as the spread of virus variants with increased transmissibility like Delta, public health measures that were applied at different times, and healthcare capacity.
There's a more detailed analysis/refutation at that link (based on vaccinated vs unvaccinated from a few countries), but here I'll provide the part most obvious to me.
Beattie essentially attributes deaths due to the spread of Covid-19 to the administration of vaccines themselves. This is most obvious when considering he calculates the highest increases for countries like Seychelles and Mongolia. Beattie attributes:
Seychelles: +10680% Vaccine Causal Impact on Total Deaths Per Million
Mongolia: +19015% Vaccine Causal Impact on Total Deaths Per Million
Arguably the spread of Covid-19 wasn't that well prevented by the Chinese (and Russian) vaccines compared to other vaccines used in the West. As the case of Seychelles would tell you, where they almost exclusively used Chinese ones. Mongolia also used some Russian ones, but saw high spread among the vaccinated too. But even these vaccines hardly caused any deaths themselves, from data that does distinguish between death by actual cause and vaccination status.
Even in Mongolia, if you compare deaths among the vaccinated and unvaccinated in the same time period, you get totally different results than Beattie's, e.g. in July 2021 [when I could find a snapshot]:
Mongolia has fully vaccinated 53% of its population, with 80% of those people receiving Sinopharm, according to Enkhsaihan Lkhagvasuren, the Ministry of Health’s head of public health policy implementation. A fifth of Mongolia’s Covid-19 cases have been fully vaccinated, but 96% of Covid-19 deaths were in people who were either unvaccinated or had received just one dose, Lkhagvasuren said.
But Beattie's analysis attributes such deaths to the vaccine anyhow. He also considers any Covid-19 vaccine (mRNA or not) to be "gene therapy". So this is double cluelessness/dishonesty, if you like.
The analysis began with an inaccurate assumption by labeling [all] COVID-19 vaccines as “COVID-19 gene therapy injections”. [...]
(The Chinese & Russian vaccines widely administered were not mRNA-based. The vastly used Sinopharms vaccines used inactivated virus.)
In contrast Beattie finds that
China: -20% Vaccine Causal Impact on Total Deaths Per Million
which is 2nd lowest result he's got (only Vanuatu had a more negative result). He says something about possibly different batches explaining such different results between countries seemingly using the same vaccines. He doesn't seem willing to consider that quarantines/lockdowns may have had any (confounding) effect, which is another interesting/glaring omission from his model. On the other hand, Beattie is very excited about ivermectin. (China didn't use/approve ivermectin for Covid, as far as I know.)