People who believe in things like flat earth have already rejected argument from authority. Ostensibly, fact-checkers exist for the benefit of people who do trust experts but also would be convinced by a random facebook post. Maybe such people exist, but, like you say, in the 5 years (?) that fact-checkers have been implemented on social media, they've already overplayed their hand and can no longer be viewed as totally consistent with expert opinion. They are often used to manipulate political opinions and distribute propaganda.

It's a little ironic. They probably introduced fact-checkers to avoid regulation, but in an ideal world I feel the most important regulation would be expressly to prevent big social media corporations from centrally controlling information like that.

I don't think 'wrong more often than they're right' is accurate, we just hear about when they're wrong and not when they're right. I also want to say that 'ineffective' is the wrong word for vaccine's relationship to new variants. The vaccine still dramatically reduces spread, and almost always prevents hospitalization and death.