Crazy Jerome didn't want to get into it, but what the heck... I will. Here are the different aspects of the 4E healing mechanics that could be used or not used, either in part or in total. Some of these things might be more tolerable to many people, depending on which parts get used.
The Healing Surge Value
This is 1/4th of your total hit points. All healing abilities use this number for how much HP you get back, rather than a number of dice worth of HP. Proportional healing. So depending on the power level of the heal, you might get one HSV's worth of points, or you might get several HSV's worth of points. In 5E, this could be adapted in such a way that rather than Cure Serious Wounds giving a PC back 3d8 + level in HPs... it might give you 3 Healing Surge Values (i.e. 3/4th of total HP) back.
The Second Wind
Each PC having his own personal "healing spell" (for lack of a better term) that a player can give to himself. It assumes that the HP the PC recovers from the Second Wind fall into the "fatigue/morale" part of what hit points mean... not that the PC is actually seeing their wounds close a la Wolverine. 4E has it as once per fight and is usable in battle... but it could also be changed to once per day (or some other amount of time) and/or only usable between fights. Several options on the "self-healing" front.
Proponents like the idea of a player having a way to regain some HP for his character during a fight... especially when the PC is fighting by himself or the party has no access to a "healer" type of character. Opponents often are ones who lean more towards "HP are wounds", and thus "self-healing" for them becomes Wolverine-like.
The Total Healing Surges Available
This number becomes the defacto way of determining "How hurt really am I?" in 4E, away from the traditional Hit Point Total. Pre-4E, your hit points would not return without overnight/long-term rest or the use of magic, so you were considered "wounded" when you were without some part of your hit points throughout the course of the day. In 4E, you can regain hit points without magic or long-term rest... but it's now your current number of healing surges that cannot. So a PC with half his healing surges gone is "wounded", even if he has total hit points. Total Healing Surges have replaced Hit Points conceptually as the method for determining if a PC is actually hurt.
Opponents find this redefinition of Hit Points away from tradition to be the major stumbling block (if not an outright anathema) to their acceptance of the system... because it removes the idea of HP itself being physical "wounds". Having HP be practically ALL just fatigue/morale/luck doesn't sit right with how they see the combat system. Proponents oftentimes don't care about any of that, because they see combat as *so* abstract that they just don't make those connections between HP and "wounds".
Death Saves & Non-Magical Healing
If we accept for the sake of argument the 4E premise that Hit Points are not actually physical "wounds", but rather are pool of fatigue/morale/luck points... non-magical healing can make a bit of sense. A warlord could inspire someone to shake off the cobwebs and get back into the fight (i.e. - spend a healing surge to regain hit points).
The disconnect comes then with the idea that someone could actually DIE as a result of an empty pool of fatigue/morale/luck. Once someone drops to 0 fatigue/morale/luck... they then make Death Saves each round and if they fail three, the PC dies. Which on the face of it by definition does not seemingly make any sense... because a PC isn't INJURED at that point, they are just exhausted or out of luck. And this is true unless you as the player are willing to go through the effort to create for yourself an explanation of what those death saves actually represent (like that that final failed death save is some monster coming over and delivering a "death blow" for example.) However, the narrative of the game itself does not define this for you, and thus there is a disconnect in the fiction of the game between what Hit Points represent, and what Death Saves represent.
So there are a couple questions that have to be asked here. In terms of "dying" after dropping below 0 hit points... which is more preferable? Losing a set number of HP each round until the PC reaches some death number (whatever you set this "death" number to be-- Negative 10, Negative CON, Negative 'Bloodied' value)... or making a 4E Saving throw each round and failing 3? Is one easier mechanically to keep track of? Does one have a better fictional narrative to hang on the mechanics?
And does the ability to counteract this death during combat by the use of non-magical healing automatically create a disconnect in the fiction? Should non-magical healing in combat be able to accomplish this? If not, what needs to happen to the system's mechanics and fiction to allow for parties that do not have a "magical healer" to save someone from bleeding out and dying? Do we WANT a non-magical healer to be able to do that?
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These are just several issues that make up the totality of the "healing surge" system. And it's definitely the case that you can deconstruct the system into its individual parts and pick and choose to use some that might work for most people, and some which won't.
But at the end of the day... my own personal belief of the whole issue comes down to one specific thing that is the dividing line and the cause for all the redefining of what damage is, what healing is, what hit points are, what death is, etc.:
Should the Cleric be the only truly capable "healer" in the game, or should there be others that could take its place within a party?
Because if the answer is that there should be others... then it opens the Pandora's box of one of them potentially being a "non-magical" healer, and then you have to jump back in to redefine (or crystalize) what hit points/damage/healing really are, and alienating players who don't like the new definition.
