Note that it can be discovered during a game that a move several turns ago was actually illegal, and the board should be returned to that position (and times adjusted). I had this happen to me once.
Additionally, there are many contradictory rules that can't all be applied, so there is a lot of room for the tournament director to apply the rules according to the spirit of the game rather than the word of the law.
https://www.chessgames.com/perl/chessgame?gid=2024861.
White castled short on move 10 and long on move 33. That didn’t help, though. White lost the game.
Also, it seems that site’s software doesn’t allow entering the full game.
Resolving irregularities does require your opponent to notice the irregularity and bring it to the attention of the referee.
Not sure about Blitz.