The problem is that not all anti climate change materials nor all anti mask material are false. There are plenty of legitimate studies on those sides, fact checking them away would be lying. So your fact checkers would need to be way more competent than your average journalist, so scaling that up to moderate the whole worlds communication is simply not possible.
But if this job is hard already for professional journalists on a full-time job, how exactly is an average person with no background knowledge and limited time and attention supposed to do it?
To me this argument sounds like "Sometimes life vests are faulty, therefore they provide a false sense of security, therefore we should get rid of all life vests and just hope people have good swimming skills".
> There are plenty of legitimate studies on those sides, fact checking them away would be lying.
If they are legitimate, why would a fact checker say otherwise?
I wouldn't know without examining the studies who to trust more there. Harvard might be getting money from some fancy "non-profit" to put out some dubious study as well.
Agreed if it were just between those two parties, one wanted to sell me a questionable product, and the other wanted me to be informed about the questionable product.