To allude to Dungeonman's terrific posts above, I think these are the points were genre-blindnes/lack of self-awareness kicks in: the players know that genre tropes will be supported (because they have the hp information to tell them) but the PCs don't.
Thought experiment if genre-blindnes/lack of self-awareness did NOT kick in:
Heroes around the land with a modicum of intelligence and self-awareness realize that they're better than regular folk, and that there are universal laws that apparently apply only to their kind.
They know that when they go into battle in full health, rarely or never is anyone taken down early in the fight; but rather through attrition. This is predictable enough that they can plan their tactics accordingly. Some adventurers can even predict that they'll never, ever be bloodied right away. Other adventurers can accurately predict that the first wound from an axe swing is never dangerous but a dozen axe wounds are.
They can ably predict that their opponents rarely or never outclass them, or they will rarely or never be put in a position to fight opponents that would outclass them. They remember goblins being worthy opponents at the onset, and after a year or so, they're taking on dragons and demons, as if they're getting better and better.
In comparison, they see that regular folk are prone at any time to a knockout or deadly strike regardless of their state of health, and thus cannot predict the ebb and flow of combat, plus these regular folk are scared of having to randomly confront an opponent who can best them and kill them dead.
The adventurers know they didn't do anything in particular to earn this destiny. They didn't bath in a magic pool of immortality or have a special sword that empowers them so. The only thing they do differently is to take on quests that others would consider foolhardy, and if they survive, they get better at it. They even get better at other skills that have nothing to do with the quest, and they can get better at these skills faster than regular folk who practice and train at it for weeks or months or years.
At first, this wonderous phenomenon is kept secret, too embarrasing and too crazy to confess. But after a few weeks, the elephant in the room is so obvious,they hesitantly share their musings. It starts with sitting by the fire at nights and discussions about luck and destiny. Then it morphs into meetings at adventuring halls, drunken boasts in pubs, and ballads sung by minstrels: stories about Karma that adventurers earn and regular folk lack.
Eventually, the Adventuring Industry is born: "Become an Adventurer! Risk your Life for Fame and Glory.. and mostly importantly... Karma! Do you really want to be a Regular guy in the King's army, or do you want to be a Warrior with more Karma after a few days in an Official Credited Adventuring Dungeon? Do you really want to spend a decade in a tower learning that Fireball spell, or do you want to unlock your very own Fireball after just a few weeks shooting Magic Missiles with your fellow Adventurers? Stop being a regular person, stop waiting for other Adventurers to save your town. Yes, you can be an Adventurer today, and get yourself some Karma! Find any Official Adventure Recruitment Centre, and find out if you qualify! (If you don't qualify, please see our Henchman Wanted section.)"
EDIT: Not to mock anyone's rp style, of course. This was a thought experiment of what it could mean for a PC to "know" their hit points. As much as I'd love to see a D&D system that encourages that...