I'm cynical, but when I see anything prefaced with "Fact checkers say..." I take that to mean what follows is probably a lie. And probably one told to harm life, liberty, or property in some massive way.
There may be nothing they could do to stifle the opinions of others that would make me more suspicious. It's on the order of "Government experts claim...".
I've seen some of the things "fact checkers" say. They'll take a tweet or story about someone's doubt, and they'll "check" things that were not a part of the piece they want you to be suspicious of. It's as though they are hoping you can't tell they've baited and switched, but will see the warning and that will be enough to convince you the writer has no credibility without looking into it.
Of course, sometimes it backfires on them.
The tweet at the top of this post was posted well before the "fact checkers say" lie hit social media. So, that's not the specific warning they applied to this one, but it didn't really work out well for TPTB. This tweet is still being shared on a massive scale. If the clowns at Twutter had ignored it, it would have been quickly forgotten. But that's not how it turned out. It's probably on course to become the most retweeted tweet ever, even though you have to jump through hoops to retweet it.
And, if you'll notice, the clowns of Twutter, after claiming that the tweet is "misleading" (it's not), want you to follow their link to "learn more about how vaccines work" even though that has absolutely nothing to do with the content of the tweet. They can't even criticize it honestly.
I see the same sorts of things happen with any skepticism concerning the most recent president selection, too. "Fact checkers say" all sorts of things that are demonstrably not true, and usually have little or nothing to do with the information they don't want you to know.
When you see "fact checkers say" you can probably ignore what comes after. It's most likely going to be an authoritarian lie. At least, if you see that preface, take anything after it with a grain of salt.
(It's probably a coincidence, but as I was trying to post this I suddenly started getting a warning that the "update failed"-- it wouldn't auto-save. That hasn't ever happened, other than a momentary glitch. But this time, it kept going on this way for a long time, over and over. It's enough to make a person suspicious.)