My own view is that hit points straddle a line between serving the sort of representational function that you describe, and being a purely metagame device.hit points - a thing only known to the player - correspond to things which are only known to the character in such a way that the player using hit points to make a decision is just facilitating the character making their decision with what information they actually have available.
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ignoring that hit points are the means by which the player is informed of a variety of things which the character can observe that the player cannot is a thing which DMs that are trying to avoid "meta-gaming" <snippage> seem to do intentionally. Having not asked any such DMs if they intentionally pretend that not only can a character not see the number of hit points they have, but they also have no clue at all about the conditions which hit points represent such as not knowing that they are feeling worn out, bruised, beaten, or so on, I use the word seem because that is as accurate as I can be.
There's no doubt that characters can (in the fiction) feel tired, forlorn, etc, and the way that the player acquires awareness of this is via hit point depletion.
But hit point depletion can also correlate to a loss of resolve, and to ablation of luck and divine protections (Gygax emphasises these things in his AD&D rulebooks). I'm not sure that the character is necessarily aware of these things. In my own life, I know that I'm not always the best judge of when my resolve is failing, or of when I need bucking up.
In my 4e game, when the player of the cleric or paladin asks "Does anyone need healing?" and the players all report on their hit point status, what I envisage happening in the fiction that the character is looking around to see if any of his/her comrades appears to be flagging or looking hard-pressed. S/he then speaks a word of benediction (Healing Word, Word of Vigour), perhaps while gently touching the ally (Lay on Hands), and the character's spirits are raised and s/he is able to go on.
I don't have such a strong mental image for what happens in AD&D or 5e, which seem to use more of a "cure wounds" model and which also require the cleric or paladin to choose how much healing to do (level of slot; or hit points of LoH) - which sort of implies that high level characters (like Conan) have gigantic melancholies and gigantic mirth, and hence need more powerful magic in order to be roused when their spirits flag.