My preferred version would require a heavy retool of the rest of the system. But here it is.
First, we'd have HP that replenishes with a 5 minute rest, where Hit Points means "ability to survive an attack that hits."
But on top of that you have wounds that can occur from critical hits. And if you're out of HP, every attack that hits causes a wound. Usually someone who drops to 0 HP just gives up. (And if you score a critical hit against someone with no HP, they
Wounds persist until you get treatment, which might take an hour, a day, or a week (or magic!). The location is random, or you can spend a bonus action to aim at a specific location, but aiming only matters if you inflict a wound. Arm wounds make wielding stuff harder. Leg wounds make moving harder. Head wounds impede your senses or might knock you unconscious. And torso wounds cause blood loss (so you make a save each turn to avoid passing out) and lower your max HP, so when you take a rest you might only get back to 3/4, 1/2, or 1/4 your total.
Wounds can be light (heal in an hour), moderate (heal in a day), serious (heal in a week), or critical (never heal).
When you drop to 0 HP, you automatically get a moderate wound wherever the attack hit, or a light wound if you choose to fall unconscious.
Normally critical hits would inflict serious wounds. They'd upgrade to critical if you have no HP left. Whenever you take a wound, you can make a Con save to reduce it by one step. And you'd have hero points you could expend to reduce any wound to light.
A 5-minute Medicine check (DC 10) can let the patient recover from a light wound. So can a cure light wounds spell (1st level).
A 1-hour Medicine check (DC 15) can let a patient ignore the effects of a moderate wound, though the benefit goes away after the character falls below half HP. Cure moderate wounds (2nd level) can fully recover a moderate wound.
Cure serious wounds (3rd level) can fix a serious wound. Cure critical (4th level) can fix a critical wound, which includes regrowing severed limbs. Then Raise Dead (5th level) lets you beat death, but any wounds the person had when they died require an extra cure spell. Heal (6th level) can fix all your wounds at once. Resurrection (7th level) can restore the dead and fix all your wounds. And True Resurrection (9th level) is that, but with a casting time of one action.
None of these actually recover hit points. For that you either need inspiration by an ally, or you can take an action to get a second wind, or maybe a heroism spell.
The idea here is to have rules for how wounds actually function in real life, but in a way that players can still function and be heroic while injured.