That's why I like fate/fortune as the backdrop, rather than merely 'dodge points'--surely, a blessing from the gods is capable of restoring one's pool of good fortune?

Well if the avoidance is skinned as things like, the blade is turned and instead hits you with the flat, leaving a bruise, or you get your blade in the way enough to slow but not stop the momentum so you receive a small cut, they work the same as they do now. While a person can ignore small injuries during a fight and maybe for one or two fights after, eventually they will wear you down.
I would use it as more an energy restorative. After a fight, even if you avoided being hit for most of it, you will be exhausted if moving in armor, carrying large backpacks, and swinging metal weapons. Avoiding damage will take a toll on your ability to move in the future, making you a half step slower. Not enough to cause you to be unable to fight but maybe that near miss can't be quite avoided next time. Cure wounds will turn into a Gatorade bottle or a 5 hour energy shot essentially. Giving you the burst you need to make it through the next fight

Yes, but somehow (to me) it just doesn't feel as 'real' as tending a bleeding stab wound or a bite oozing with venom. In a game where the role of "healer" is so well established, it seems like this robs that character of one of his main functions, replacing it with a kind of mystic ATM role instead.

I won't deny that there's an alteration, but divine magic also is responsible for most of the other, more physical healing magic, as well (Cure Disease, Slow/Cure Poison, Restoration, Raise Dead and so on). To my way of thinking, the notion of them restoring their companion's good fortune is actually more... thematic, I guess? to their being the agent of their god than having them serve as walking First Aid kits. Of course, really, the spell names need to change for this to hold well--Cure Light Wounds to something like Lesser Aid Destiny, or somesuch.

Furthermore, that means the characters are aware of their "luck pool." I think that changes the magic of the world, even if only in a pure fluff sense, which will have consequences on the way PC's and NPC's act and treat the world. The idea that everyone has a temporarily finite amount of luck and that they can use it to do things, like survive jumping off a cliff, creates a very interesting world, but one that feels like it is significantly different from traditional D&D ones.

Good point.

I would find that harder to get into the mindset; the suspension of disbelief, whatever you want to call it. It's too abstract. It would almost be like your character actually carrying dice around and rolling them to perform ability checks.

Fair enough--it's a notion, not a One True Method. For me, suspension of disbelief is harder with the traditional interpretation of hit points as physical injury only, because of how that plays out (such as the aforementioned instance where someone knows he can survive a jump off a cliff because of his hit point total rather than believing himself protected by destiny). It's obviously gonna be a personal thing one way or the other.

Cubicle 7 did that for their Dr. Who RPG. You start the game with 12 "story points" which are effectively your HP in the game. If you run out of Story Points, you either take damage to your stats, or if you're facing a Dalek, you die.

Try this:
Hit points are a combination of fatigue and luck and even some skill. Once those are whittled away, you're not dead, just vulnerable. At that point you make saves vs. dying when struck to avoid injury.

I don't think that's necessarily the case. We're assuming that the character is making a conscious choice in to what spell they cast, and we don't need that - the player is doing so, but for all the character knows they are just annointing their companions with holy oil or whatever like they do after every potiential disaster averted by their god. That this happens to restore the luck pool is incidental, and it doesn't even have to - they can annoint, but the player can choose not to spend spells on it and f or it to not actually do anything.

In short, the players are aware of their luck pool, but the characters don't have to be at all, which makes suspension of disbelief easier.