But if hit dice are recovered at 1 (or 2, etc) per day, rather than all at once, natural healing will exhibit the problem.
I agree with that, but this one of those things were my appreciation for tight, predictiable mechanics and natural sympathy for the old-school "wearing down of the party over time" are at war.
A rather interesting side effect of this mechanic, if I read it correctly, is that early in an "adventure" the party will function a bit more like a 4E party. They are topped out, healed up, ready to go. However, as they adventure continues, hit dice are used, and not fully recovered, the party moves more and more into functioning a bit like an AD&D party. Magical healing becomes more and more critical to keep going. Of course, this is relative, as when I say "bit" I mean that shift will be as extreme as straight 4E or AD&D.
That, in turn, has rather interesting implications for house rules and/or modules. If you want something more like 4E, you can be generous with the restoration of "hit dice" explicitly, use short "adventures," or arbitrarily make a bigger "adventure" into a several shorter ones. It all ends up the same, as people get chances to heal up fully.
OTOH, if you want something more like AD&D, you can be rather strict with "hit dice" restoration, deliberately define "adventure" as somewhat longer than normal, or even set up environmental factors. (For example, you could put the "adventure" out in the wilderness, a week from civilization, and then knock off a few "hit dice" for fatigue before the adventure technically starts--whether you play that out with skill checks, wandering monsters, or start the party at the dungeon door with fewer "hit dice" as an abstraction.)
Heh, this brings to mind the Dragon Quest "fatigue" mechanic where the game master is specifically instructed (with some detailed rules) on how to adjust starting fatigue based on the circumstances. You are in better shape when jumped in the tavern in town, after resting for a few days, than you are the end of a long day travel in the wilderness.