It always struck me as odd that the checkmate definition should be so (relatively) complicated. A much simpler definition is "you lose if the opponent captures your king". A checkmate is just a situation where you can't avoid getting your king captured. If someone doesn't notice their king is on check, then they could lose the next move. After you pass the beginner phase, this will almost never happen to you anyway, so most games will end the same way: checkmate or losing because of time.

When I was in school, we used to play blitz like that. You lost track of your king, boom, you're dead