In fact, this is one of the easiest scenarios for a fact checker to handle correctly.
The source is a reputable, top-tier peer-reviewed publication. Even an inexperienced (but competent) fact checker could check run literature searches on the journal, reviewers and authors. This would reveal that they’re working at reputable institutions, and have been for decades.
It’s not hard to keep a list of journals and assign them a reputation score. Commercial services do exactly this.
But even the best journals screw up, especially in their blog posts!
Remember the “fact-check” wasn’t for the peer reviewed article itself, but a blog post promoting it.
Even if the BMJ has a high reputation score, not all of its blogs should share that. It might be run by the social media team - not the editorial staff.
That's why the fact check report doesn't mention any specific factual inaccuracies. Context matters, missing context is a problem.
EDIT: see here https://news.ycombinator.com/item?id=29597047
Even though the medical journal article accurately shows problems with this specific aspect of the medical/clinical review of the vaccine, the vaccine is proving effective in combating the transmission and severity of the covid virus. Many of the people currently going to the hospital for covid are unvaccinated. This is placing a heavy burden on the hospitals and preventing other people from getting the care they need.
People are fighting over specific facts because they are trying to influence the broader narrative/discourse going on in society about covid, vaccinations, lockdowns, etc.
Comments in this thread are discussing facebook, journalists and fact-checkers, and their roles in influencing public opinion and social discourse.