Heh. That tautology took longer to pop up than usual.

Yes, hps are a countdown to death, that that is their only function, and that hp-loss "wounds" do not in any other way impair you, is one of many 'realism' issues with them.
One can assume that if lower hit points indicate that you are less likely to avoid a killing blow, then you can also assume that the character is getting tired, sloppy, out of breath, run-down, sweaty, his pulse pounding in his temples.
Loosing hit points is less like gaining a wound and more like being worn down from the fight.
I don't know about Fifth Edition, but Third Edition has Feats where the character continue to operate at -1 HP and beyond. Some 3E d20 versions have special rules that allow a character to continue functioning at a reduced level at 0 HP and sometimes beyond.
In 1E AD&D, Gygax speaks of characters continuing untill about -3 HP or so.
Yes, it's all abstract, but "conditions" can be assigned to a character based on how long it takes the character to heal.
In the d20 3.5 based Conan game:
Losing any HP, but above 1 HP? The character is being weathered down in a fight. His endurance is fading. He's less likely to avoid a killing blow (or a blow that does serious damage). Damage Points heal relatively quickly, so much of this type of damage is just resting up (sore muscles, maybe a bruise or a light cut).
At 0HP, the character is staggered. His actions are halved. His wound is akin to the above, but he's been brained pretty good. Or, he's taken a blow that makes him a lot less capable than he was before. But, he'll heal relatively quickly.
At -1 HP, the character takes a wound that drops him (unless the character has a special Feat that allows him to fight on in spite of this wound). This may not be a life-threatening wound, but it's enough to knock him out of the action. Maybe a blow to the head. He'll heal much, much slower.
At -10 HP, the character takes enough damage to kill him. Maybe he laid on the ground and bled from a wound. Maybe he was rammed through with a yard of steel. Maybe his head was chopped off. He's dead.