For the past few weeks Legends & Lore has brought up the issues of healing, first by asking if removing the cleric as a necessary element of play was something that they should be doing (Yes!), then by pitching an adjustable-yet-flat healing rate (No!).
Both 2nd and 3rd Edition have pretty much always assumed that you had a cleric, or someone else capable of casting cure etc wounds spells multiple times per day (which in 3rd Edition could be summed up as either a favored soul, or the stop-gap approach of "guy with lots of healing wands"). Otherwise you had to rely on natural healing which, to put it nicely, was slow: if you wanted to keep adventuring you got back one-a-day, but if you stayed in bed recovered either three per day in 2nd Edition (I think, might have been two), or one-per-level in 3rd.
The major downside, similar to wizards running out of spells, is that running out of healing magic effectively puts a stop to the adventuring day, unless you have access to a bunch of magic items that can heal you. I find it strange that for so long, the party was often reliant on one person to keep everyone up and running, especially when a single die roll could kill your character: I ran a short-lived Age of Worms campaign sans cleric, and my players did not enjoy burning through a huge chunk of cash for a wand that only the bard could use, which unfortunately required that she be conscious (which was a hard criteria to meet when you have an AC of 14 and only 8 hit points).
Another issue is the expectation that you will have access to healing magic, regardless of what kind of cleric you were wanting to play, which was lessened when 3rd Edition allowed clerics to swap out spells for healing magic as long as you were either Good, or Neutral and opted to channel positive energy (and even more so in 4th, where every cleric had access to a few encounter-based heals). For some clerics--Healing, Protection, even War--I can buy it, but what about those that want to worship a trickster, or a god of knowledge or crafting? Why do they have automatic access to healing magic?
For several reasons 4th Edition made it very possible to adventure without having a cleric--or really, any kind of healing class--on hand. The biggest step was by giving every character access to a measure of self-healing in the form of healing surges. The amount that each character had access to depended on your class and Constitution modifier, but since they always restored a quarter of your total hit points, fighter-types tended to have more, and benefit more from them.
Another, more controversial change, was that a long rest topped you off in every regard: hit points, healing surges, and per-day abilities. For better or worse this meant that, aside from the odd disease, that after a good night's sleep you were ready to head back into the dungeon. Finally, there were plenty of other, equally serviceable classes that could substitute for a cleric, each with their own theme and playstyle.
As of the eight playtest packet, the default healing rules in 5th Edition are kind of in the middle: you get a number of Hit Dice equal to your level, which vary depending on your class and Constitution modifier, so at the least fighters tend to recover more hit points than wizards. While you regain everything after a long rest, you do not get many (again, the number equals your level), so most of your healing depends on how many healing spells the cleric has access to.
I think that the best approach is to build a system where there is no assumed healer or access to certain magic items, but the party can keep on adventuring without having to spend several days taking it easy. By not requiring players to pick certain abilities within certain classes, it provides a lot of freedom on both sides of the screen: Dungeon Masters can run low-or-no magic campaigns, hand out the items they want in high magic campaigns, and worry less about time-sensitive adventure objectives. Cleric players can worship and pray for whatever spells they want, without having to worry about making the wrong choice or running out of certain spells.
3rd Edition's Unearthed Arcana has a lot of hit point sub-systems that can be mined, in particular Vitality and Wound Points: though more complex than just having a single pool of hit points, you could make effects that apply to only one pool. Vitality points would be based on your class and represent how hit points can let you evade blows, or turn an otherwise lethal blow into a minor wound, while Wound points would be derived from your Constitution score and represent your overall physical toughness.
The refocus action (and warlord powers) might only restore Vitality points, and they could completely fill up after combat is over as part of the whole short rest routine. Wound points would recover more slowly (at an hourly or daily rate, ideally based in some way on your Constitution), but could be restored using magic, or maybe on a limited basis using healing items and skills.
I have said before that being able to make a useful fighter/wizard early on is part of my litmus test for liking a Dungeons & Dragons game. Being able to make a viable adventuring party using any iteration of the hit point and healing rules is another. I get that they want to make the "basic" setting of the game, well, basic, but I think that people can handle fairly simple mechanics like Hit Dice/healing surges/reserve points. At the least I would like one set of healing rules that lets the party get away without a cleric, but does not see a massive power boost by the inclusion of one.