Just random thoughts.
Lasting damage is a quite different thing. And that is something D&D definitely does not model well, at all.
This is kind of the point I was making about non-lethal a little earlier on, but it applies to HP as well. I think with the dawn of a new edition they COULD and more importantly SHOULD define HP well for once. I'm not saying they should radically change how HP work or anything, or use alternate systems. But they could use different methods of how HP are gained, recovered and spent based on a deeper understanding of what they are, if they bothered putting the time into it.
Part of this is defining the issues with previous editions; why something worked or didn't work. This inspection should go beyond the mechanic but delve into the idea and concept and why those things worked or didn't work. It needs to look at the other mechanics that go along with it and why they work or don't work.
I'm currently working on my own system and I decided early on that I didn't want HP bloat of 3e and 4e. Having 100 HP and surviving falling off cliffs didn't interest me. How did I remedy this? I took a look at how characters got HP in the first place, what the parts of HP represented toward the total and then how to scale them to achieve the numbers I wanted.
I decided in the end to go with a very very different approach to classic 3e (and I assume from what we'll end up with in 5e). I'm using HP total to be full HD + CON mod + Level. That is it. Weapons, variants, crits, optional rules and most importantly damage all reflect this shift. I could institute what is being suggested by Mike Mearls, and give my fighters their HD (or HD*LVL) every day to heal up for free, but I don't find it necessary. The 3.5 version (for which I am using the OGL) specifies that characters get their level or HD per day works fine for natural healing and for my games it makes much more sense.
I think that if they did something similar, and re-examined the root concepts, that they would end up with a superior system instead of giving us the same assumed stats (to gain HP) as before with a new natural healing system.
And critting gets even more meaningless when(using Pathfinder here) a Fighter can crit on a 10 or less.
How do you crit on a 10 in PF?
No moping about in bed for days. No magical healing potions (just a few ibuprofen), and she's up and moving about and working, and way more worried about her appearance than she is about anything else.
EDIT - And if she was a higher level fighter, or had some martial arts training, she could maybe have rolled with it, and minimized that damage a lot further (perhaps even having no damage at all).
EDIT 2 - All this, just to note that lower-level characters bouncing back to full HP faster than trained warriors makes a sort of sense. Less damage is required to hurt them, so there's less damage that needs to be healed.
First, ibuprofen effectively IS healing potions, at least for "pain" damage.
Second, your first edit talks about reducing damage, not changing the meaning of damage. It is a mechanic related to damage reduction instead of HP.
Third, for your second edit, I've always understood HP to be less about "bouncing back" to normal and more about bouncing back to a healthy. On the first count, yes full HP would be important. On the second count "back to normal" one could assume 1 HP, or 10, or bloodied or full or anywhere in between to be "healthy". Only with Bloodied do you have the distinction that someone is unhealthy below 1/2 HP.