It's good to see that the vast majority are in favor to at least some form of limited self-healing. Personally I had grown tired of relying on the casters to keep the game going last edition, and 4th Edition made it easier--I would daresay possible--to manage without a healer at all, or at least without having to burn through cash stocking up on healing potions and wands (the latter of which required characters to spend ranks on Use Magic Device).
The thing is that if hit points do not--and never did--simulate physical trauma, then there is no reason to severely cripple their "natural" recovery rate. 2nd Edition did this at 1 point per day, though 3rd Edition allowed you to get away with 1 per level, so long as you got a full day's rest. A Heal check might have been involved, as well.
Healing surges provide a nice, solid mechanic for giving players an idea of how much more punishment they can endure, as well as providing a resource that could feasibly be used for other things, such as fueling rituals and/or abilities. For example, what if sorcerers got at-will spells, but no encounter or daily magic, and instead had to spend healing surges. This could work for other spellcasters, representing them becoming exhausted. It would also work for martial types.
Some players claim that healing surges can strip out the drama of a situation, since players can crawl out of a pile of rubble and "magically full-heal", but this does not seem much different from just having a cleric spam cure whatever wounds until the character is topped off (or drinking a bunch of potions). Heck, give her a few levels and she can just conjure up free food and water, and also instantaneously alleviate ability score damage and level drain (assuming the characters did not burn cash on items that can do all that, anyway).
Things that might help is preventing characters from using all the surges that they want. After a battle characters might be able to spend just one, or could only spend enough to get to a certain percentage (say, bloodied value or 3/4s or something). An extended rest would allow them to exceed this amount, but to avoid a lengthy nap resetting all their stats to full, you could also cap the recovery rate. So, fighters might get 4 + Con modifier back, wizards 2 + Con modifier, or characters might only get back their Con modifier (so tough characters still recover faster). This way characters would still want to be stingy with their hit points and not just blow through surges willy nilly.
There could also be rules for extended injuries, so characters could suffer penalties for taking critical hits (perhaps in lieu of extra damage, similar to that mechanic in Dresden Files that lets you keep going, but at a cost). Things that healing surges cannot fix. I think this would work for grittier games, even going so far as specifying what sort of injuries that curing spells can fix (something I remember from 2nd Edition). The balancing act would be making it so that an injury is cumbersome, but not adventure-stopping. I would also like to avoid the whole ability damage fiasco of 3rd Edition; just give me flat penalties so I do not have to go back and re-factor my attacks, damage, skills, etc.