> Once you create a fact checking entity, it will decide for itself what are object truths

okay, so what? What do you think a court does? The guy who reads the meter at the water sanitation plant or someone who hands out parking tickets?

A fact checker is nothing but an institution with the authority to adjudicate on some questions which are relevant for the maintenance of the platform, no different from any other authority that manages public conduct. Where is the problem?

And of course fact-checkers are not unaccountable. Like in this case their behavior itself is topic of public discourse.