Interesting idea Kirk. I think 4E has something called "healing surges" which might be similar to what you are proposing. I am not sure, since I have never read 4E.
Your idea hits upon one of the "debates" in D&D, which is "are HP representative mostly of bone and muscle, or are they mostly stamina?". If they are mostly stamina, then most of them should restore fairly rapidly. So one might "heal" 2x level if a front line fighter and 1xlevel if not, after every encounter, just as a quick and dirty rejuvenation.
Certainly, if HP are mostly stamina and luck, one would expect most would restore naturaly after one night's sleep. Thus, the typical resting healing rate of 1 hp/day seems completely at odds with the "mostly stamina" view of HP.
One problem with the stamina view, and the subsequent easy healing that results, is that the balance for dungeons can get a bit out of whack, since dungeons are often built more on the "long resting" assumption, with magic being the special shortcut in the dungeon.
Some games also have a bifurcated system where damage is counted against stamina first, but then eventually turns into wounds, often with some negative effect on character performance. I find these models excellent in theory, but in play they always seem so much more fiddly and book keepy than good old D&D where you just track HPs, and try to ignore the fact that the 99 HP fighter fights exactly as well when reduced to 3 HP, and without magical help can be expected to spend a few months resting until he is back to 99 HP