I think "fact checking" has lost its legitimacy in the eyes of many people — and journalism in general. You can't convince people that something is objective when it has an obvious bias.
Granted, we know that some beliefs are incredibly resistant to contradicting evidence, and can even result in people doubling-down (e.g., anti-vax). (Ironically, I don't have a citation for this at hand.)
The fact that someone can gain credibility by announcing that they are a truth teller explains way too much about the reliability of garden variety human empiricism.
“Fact-checker” is just a sad last-ditch attempt to salvage this “Arbiter of Truth” status from the smoldering remains of the once-respected “journalist” profession.
Think of The Washinton Post's absurd "fact check" of Bernie Sanders. https://www.rollingstone.com/politics/politics-news/bernie-s...
Granted, we know that some beliefs are incredibly resistant to contradicting evidence, and can even result in people doubling-down (e.g., anti-vax). (Ironically, I don't have a citation for this at hand.)