Facts are facts, and nothing a human says is a fact, it's a projection of a fact upon their conscience, at best.

And those doing the "fact checking" are humans, so they are checking if something is fact in their own opinion or organization's policy, at best.

These are truisms.

There is no rejection of fact checking that will result in more truths being exposed to the world, only less.

This is wrong. People like to pick "their" side in power games between mighty adversaries, and to think that when one of the sides is more lucky, it's them who's winning. But no, it's not them. If somebody's "checking facts" for you and you like it, you've already lost. Same thing, of course, if you trust some "community evaluations" or that there's truth that can be learned so cheaply, by going online and reading something.