Would you say the apparent contradiction between what you and other commenters are saying is partly explained by the high volume of games you're playing? Or do you think there is some other reason?

I wouldn't. I never progressed beyond chess clubs in public schools and I certainly remember people making illegal moves in tournaments. Like that's why they make you both record all the moves. Because people make mistakes. Though, honestly, I remember more notation errors than play errors.

Accidentally moving into check is probably the most common illegal move. Castling though check is surprisingly common, too. Actually moving a piece incorrectly is fairly rare, though. I remember one tournament where one of the matches ended in a DQ because one of the players had two white bishops.