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.

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.)