I'm perfectly content with abstract HP and I strongly dislike Healing Surges.
I think the difference is presented in the subtle shift from here:
Even though HP was always meant to be an abstract gestalt of different factors--body, vitality, endurance, even luck, etc--...
To here:
At least as long as HP remain HP--that is, a gestalt of different factors, an abstraction.
Yes, it was always an abstract. But it was understood that physical damage was the benchmark of that abstraction. Setting magic aside, healing was tied to ideas of normal healing of physical damage. Yes, it was a loose tie, but it was there. And along with the idea that some HP represents "luck", "fate", raw skill at avoiding damage, whatever, it was simply accepted that recovery of this "luck" scaled consistently with the primary idea of physical healing.
And that had a benefit because, at least, it kept the abstract part of the healing concept on the same side of the ledger as the abstract side of the HP concept.
But now it has gone away from being an abstraction that is benchmarked against physical damage and healing and into being a free floating abstraction. There is no connection to physical. And so you get a situation in which two warriors fighting with weapons end with both or either of them either dead or able to shrug off any and all damage with no meaningful recovery period or medical aid.
When you move away from an abstraction built on a physical idea and to a pure abstraction, no damage is physical at all. No damage may ever be dealt that con not be abstracted away in a full absence of anything resembling actual healing.
Now, you add magic into the system. As long as magic healing still scales with this abstraction, it still works good enough. It doesn't really matter if 15 points of healing is closing a wound or restoring luck. So that in itself does not change anything.
It has been pointed out that the wand'o'heals creates the exact same game result, so what difference does it make. My first response to that it is not correct to assume this happens in everyone's game, so it certainly isn't correct to assume that designing a new game expecting everyone to accept it is ok. But, even when a wand is used in rapid succession, at least there is an implicit narrative justification for the healing. Using "magic" to channel power and heal wounds is a fundamental idea of how a world works. Joe the fighter can close his wounds through willpower is not. Certainly you can invent a world in which everyone has a form of regeneration. And that is cool. But the default concept of a fantasy world does not resemble this idea.
Yes, you can think of specific examples in fantasy that perfectly fit a surge. Princess Bride is probably one of the best. And yet, imagine how miserably anti-climatic it would have been if everyone had already been "surging" four times a day throughout the movie prior to that.
You asked for solutions. I have an option in my game that provides exactly this kind of heroic moment, and would readily be recognizable as a form of "surge" but with two key changes.
First, it requires an action point, which is a much more precious resource. The effect is not something that happens routinely, but only at desperate times. Second, it provides temporary hit points, not healing. These temp HP go away within 10 minutes and deal an extra 50% damage when they do. You can heroically fight on, but you still need to heal, and that much more for having exerted yourself beyond your normal means.
Another option is a wound/vitality system. These have there own issues and I generally don't bother with them. But if you must have surges, W/V can keep the surges fully in the abstract side of things.