FrogReaver
The most respectful and polite poster ever
Well, I don't have the first problem, because I don't have damage represent injury that requires more than a night's rest to heal, and I don't see the second problem as a problem. If I did, I'd add in something like the optional Lingering Injuries rule in the DMG and leave hit points alone.
1. They aren't recovering "unnaturally fast" if they aren't sustaining injuries that require more than an eight-hour rest to heal. As DM, you're fully in control of how severe you describe a PC's injuries to be. If you want damage to represent more serious injury, there's always the Gritty Realism option. Playing with that dial could allow you to use damage to represent injuries that take up to a week or more to heal. The Slow Natural Healing option, on the other hand, requires the expenditure of hit dice to heal at all without magic. So options are available, but theoretically you could still describe injuries so severe that a rules-based solution would be difficult. I don't think that's a problem with the rules, however.
2. I think injury has never been a feature of the game because it's considered un-fun to try to play with a PC that's only partially functional. Lethality, however, is something that provides a lot of dramatic tension to the action, so the idea is you get to play your PC all the way up until s/he's dead, and you don't have to sit on the sidelines with a bum leg or whatever.
I would tend to think the kinds of injuries you get in battle that knock you unconscious and leave in the "process" of dying tend to be severe injuries that take longer than 1 nights rest to heal. Do you agree?