A majority of Republicans believe Trump won the election (!) because that's what Trump is saying. There is little reason to think a Republican fact checker would contradict Trump. Indeed, any fact checker who did would be ostracized as a RINO, much as has happened with Fox News since they acknowledged the outcome of the vote.
Partisan fact checkers would only give the Republican party a safe space to continue to lie about matters of objective fact, such as who won the last election.
Frankly, the fact that 'what is truth' can be framed as a partisan issue in America speaks to much, much deeper societal problems than organized fact checking can fix. This is an attitude that is much more in keeping with that found in countries like Albania or North Korea rather than in another advanced democracy.