I was there, Gandalf. I was present and accounted for during more than one of the pointless and boring bickerfests about what hit points really represent. This argument is deader than Internet debates about Roe v. Wade. The people with their opinions are not going to change them, no matter how impassioned and reasoned the discourse. Your perfectly executed logic trap arguments are dumb. Hit points are:
Unrealistic! A high-level character can fall from great heights and survive.
Nonsensical! Healing effects heal wizards proportionately more than fighters.
Contradictory! Hit points don't represent wounds, yet characters still make savings throws vs. poison when struck by poisoned weapons.
Confusing! Describing hit point damage from a sword, a dragon's breath, and a magic missile are all different.
Silly! A character can be stabbed a dozen times, yet he fights as well with1 HP as he can with100 HP.
Stupid! Hit points aren't meaningfully restored by medical attention outside the realm of healing spells.
Seriously, we get it. Hit points don't make sense.
Dear dorks: Gygax already settled this matter back in 1979. From the AD&D 1e DMG.
It is quite unreasonable to assume that as a character gains levels of ability in his or her class that a corresponding gain in actual ability to sustain physical damage takes place. It is preposterous to state such an assumption, for if we are to assume that a man is killed by a sword thrust which does 4 hit points of damage, we must similarly assume that a hero could, on the average, withstand five such thrusts before being slain! Why then the increase in hit points? Because these reflect both the actual physical ability of the character to withstand damage - as indicated by constitution bonuses- and a commensurate increase in such areas as skill in combat and similar life-or-death situations, the "sixth sense" which warns the individual of some otherwise unforeseen events, sheer luck, and the fantastic provisions of magical protections and/or divine protection. Therefore, constitution affects both actual ability to withstand physical punishment hit points (physique) and the immeasurable areas which involve the sixth sense and luck (fitness).
Next line of discussion: why does my Lawful Good cleric get better at praying for murdering baby orcs?
Whenever you have a question about why hit points work the way they work, the answer is always "they're an abstraction that serves to reinforce the gameplay loop." That gameplay loop as envisioned was an endurance test. Characters enter a dungeon with a handful of hit points and try to make them last as long as possible while delving. As characters level up, their hit points increase, so they can stay in the dungeon longer and overcome greater challenges.
From a higher-level perspective, hit points are a metacurrency spent to avoid bad things happening to your character. Bandit stabs you, you die. Don't like that? Spend 1d8 hit points to avoid bleeding out while he rifles through your pocket. Angry dragon breathes fire, you die. Don't like that? Spend 10d6 hit points to avoid burning to a crisp. Village busybody tugs her braid and beats you over the head with a stick, you do what she says. Don't like that? Spend 1d6 hit points to tell her off.