For me, HP is my injury mechanic.
A guy who hits you with a sword isn't hurting your feelings, he's hurting your body. But I can't be arsed to try and figure out exactly how and why he's hurting your body, so I figure this works okay: I reduce a pool of points you have. These can be thought of as your "reasons I don't die from getting hit with a sword" points. What, exactly, that means, is kind of up to you and me in the moment, but the salient fact of the fiction is that you've been hit with a sword, so it is going to cause some bodily harm. Someone with fewer of these points WOULD die from getting hit with the sword.
So if you take damage, you take some bodily injury. How severe that injury is depends on how much HP you've got left after the fact. Is it NONE? The injury is life-threatening. Is it SOME? The injury is not life-threatening. But the precise nature of the injury doesn't matter to me. Say it's a broken leg. Say your arm's off. Say it's some bruises. Say it's some internal hemorrhaging. Whatever makes sense to you and me and everyone else works. The more badass your idea of your character is, the more he's going to be able to continue fighting with his guts as an ascot.
Which is the takeaway, here: HP is the only mechanic I need to model bodily injury. Having a broken leg or a sucking chest wound or a cosmetic slice or a meaty chunk taken out of your leg isn't something I need a lot of detailed rules about. You're down HP. You might not survive the next time you are hit with a sword. So you're going to be careful. I don't need a litany of fiddly modifiers to directly model the specific wound you're suffering, but I do need you to understand that you are WOUNDED, because you have been hit with things that could kill you...and you didn't die.
Hurt fee-fees don't make you die, and the barbarian who hits you with his greataxe isn't just hurting your "protagonism." But maybe you just sprained a shoulder instead of getting your head lopped off, and maybe you attribute that to luck or skill or morale or divine influence and that's all good, but you are still suffering a physical effect from being hit with something that would be potentially deadly. That's why you're missing HP. You've suffered and injury.
Damage on a miss doesn't bug me in that context, because a miss is just an attack that doesn't do physical damage. Maybe Barry's attacks ALWAYS do physical damage, no matter how agile or well-armored the target is.
Though in that respect, perhaps it should be a save vs. Barry's attacks rather than damage-on-a-miss?
Damage-on-a-miss also bugs me a bit from a gameplay and fictive perspective when it can be spammed ad nauseum from a low level, because it means your first-level trainee is suddenly UNAVOIDABLE, and that's also a little dull, but I buy it on a "realism" level just fine.