Then who monitors the monitoring organisation? A centralised solution to information oligopolies cannot work.

Only court can prove that something is fake or not fake. Other solutions are vulnerable.

Extreme example: Russian plane was shoot at Black sea in 2001, and Ukraine is blamed by every article in internet you can find, but not by court.

I'm 100% sure that every fact-check organization, except court, will confirm mainstream «fact».

Agreed. An important part to this is transparency of the members working on this as well. Almost every fact-checker I've read up on has a conflict of interest on many of the "facts" they're checking.