Then the people who decide what the politicians are allowed to say would themselves be politicians.

A dictatorship of politifact? No, tech firms are experimenting with that already. It results in things like tables of public data being classed as "disinformation".

The whole idea of fact checking is naive. Nobody is trustworthy enough to determine truth. Everyone who tries turns it into "truth is whatever powerful people say it is" (which is itself circular logic of course, but the people deciding on the meaning of truth are rarely all that bright).