This is an extremely arbitrary change that will break a lot of the intended functioning of the system
The system is designed with certain HP values in mind. Damage scales rapidly with level, in fact faster than HP does, and thus after only a few levels past the cap, just about everything will kill in one shot.
The system was already broken though.
Enough optimization, and most melee types can one-shot just about anything of a remotely appropriate level that they can get their hands on. This just makes doing that much (much) easier.
Ultimately... perhaps it would “work.”
But most players don’t play at those optimization levels, and most combats are not supposed to be decided by initiative, even if at the highest optimization level that’s often what it comes down to. Gentlemen’s agreements, ban lists, houserules, usually they’re designed to mitigate if not eliminate the rocket-tag nature of D&D, while this would put it into overdrive.
It would be very strange, and extremely lethal. Death would be common, and would frequently be arbitrary. Characters will often die without actually getting a turn in which to do anything to try to prevent their death. Ambushes would be almost impossible to survive; a single rogue archer getting a full-attack before anyone else has gone (i.e. they’re flat-footed), or from a hidden location, could easily kill the entire party in one turn. Initiative would be, by far, the most important combat stat (which it arguably already was), because whoever goes first, will kill the enemy before that enemy can go, most of the time.
Which doesn’t sound like an overly fun game to me, but I gather some do prefer it. As I said, at high enough optimization levels you can force the game into that mold anyway. At least this change eliminates the “skill gate” for you to get in on the super-lethal combat.
If that’s what you want, and more importantly, if you can find players who want that, maybe it could “work,” for a given definition of working. I wouldn’t play, I don’t think I know anyone who would be interested in playing it, but maybe you can find them.
But you perhaps misunderstand HP
Based on this comment:
But there are other solutions - give PCs so many free rerolls (favors from the Gods...) for each adventure (or per level), something used in Deciphers's LOTR RPG. Or be generous with magic armor & shields. Magic armor is a good proxy for extra hit points, minus all the hand-waving trying to explain how a PC can survive unbelievable amounts of damage.
I think you are misunderstanding the breadth of HP’s abstraction: it does not (necessarily) represent physical health, nor does lack of it (necessarily) represent injuries and wounds.
It also represents favors from gods, innate magic, destiny, luck, morale, and so on. Damage can be done from a lot of causes, and it can also be healed in a lot of ways (there are healing effects that derive their effect from inspiration, for instance, and some of those are extraordinary, i.e. nonmagical, in nature). HP is a large and expansive abstraction, but it’s not just “characters can soak absurd numbers of wounds and injuries.” It’s often closer to “plot armor.” Note also that loss of HP does not result in penalties due to wounding. Part of that is simplification and cinema, but part of it is also an acknowledgement of what HP is not, namely wounds and injuries.
Vastly superior option
As everyone has been saying, E6 is the way to go. E6 allows you to maintain that “low-level” feel, capping players at 6 class levels (and that includes 6th-level HP), but allows them to continue to gain XP, which they can trade in for bonus feats. It’s a great system, giving just enough to keep the game moving and the characters growing, while preventing more powerful effects that greatly change the nature of the game from getting into play.