In Chess Notation, How Is It Recorded When A Player's Clock Runs Out?

But, your ideal world already exists.
People who want to play with a clock, play with a clock. People who don’t, don’t.

Tournaments need them, because tournaments necessarily need to have clear rules and be decisive.

I suppose that one could have a per-move time limit, that reset every move. That would still force an eventual definite end to a game, but would make losses by time extremely rare, because the player would always be able to make a move, any move, before the clock ran out.