People on the right often don't even want the fact-check system to exist in the first place. It was pushed by left-wing people who hoped they could use it to kick out the right.
Talking about people like Bolsonaro, Modi, Duterte or for that matter Boris Johnson.
The thing is, on some level I think a lot of the cheerleaders of fact checking as a weapon against right-wing information must know this and want it. There's been quite a bit of outrage over organisations fact-checking left-wing claims as false, especially when it's a fact checker affiliated with a decidedly non-left wing and non-Democrat supporting site that does it, including claims that it's destroying democracy, dubious arguments about false equivalence, and using the possibility of Facebook penalties to imply it's suppressing left-wing speech.
(There was also a really revealing incident where, after a state Democrat party got caught really blatently violating Facebook's rules against voter suppression ads in a tight election, nearly all publications just didn't cover it despite thorougly covering the announcement of those rules. I don't think they got penalized beyond having the offending ads pulled, so the victim narrative would probably have been hard to pull off.)