On alternative to traditional hit points can be seen in OSR/NSR games derived from Into the Odd. The game still has HP, but it stands for "hit protection" instead of health/hit points. In Into the Odd, there are no attack rolls, you just roll damage dice. HP is then a buffer that resets after an encounter to absorb a hit or two. After that, characters and monsters start taking all damage to their strength stat, which provokes critical damage checks that can knock them out of combat.

So, the result is that combat is very fast, a couple rounds at most, and very decisive/deadly without having the classic OSR issue of your 1 HP wizard dying because they ran into a cat.