"fact-checking" is dog whistle for "only leftist allowed"
American conservatives would fall afoul of fact checking far less frequently if the conservative worldview wasn't wedded to talking points that are not factually accurate. For example:
There is no evidence of widespread voter fraud in the 2020 presidential election. There is no evidence of ongoing large-scale voter fraud in US elections. There is no evidence that the incoming Biden administration intends to end capitalism. There were no 'convoys' of migrants preparing to violently storm the US-Mexico border prior to the 2018 midterms. Barrack Obama was not ineligible to serve as President. There is no evidence that the Democratic party is detaining children in the basement of a pizza parlor. There is no evidence that the Democratic party is run by a cabal of pedophile cannibals. Etc.
A fact checker who does not point out the fact-free basis of these, and other, deeply-held conservative talking points is simply not doing his/her job.
If the weight of fact checking falls primarily against conservatives, then conservatives can fix that problem by lying less frequently.