People with poor critical reading skills will probably just accept (or, if it conflicts with the preexisting world view, reject) the original claim as true without referencing a “fact check” at all.
In my experience, people with poor critical reading skills don't read. Instead they just regurgitate what they heard on FOX, which is that fact check websites are a liberal scam.
The same thing happens on the left with the New York Times. Laziness is a trait orthogonal to political party.
New York Times standards are much higher than Fox. Are they perfect? No. But generally they are far less likely to lie to you.
https://greenwald.substack.com/p/a-court-ruled-rachel-maddow...