Handwaving inventory considerations is fine. Having a class ability that makes nonmagical items magic is not (unless it's a magic class).
Yousay it's magical. You draw the line at that point, I draw it elsewhere (at a less restrictive place that opens up more design space for creative players).
The rules are badly designed in that hps are not meat and blood.
Two things : 1) If HPs are not an abstraction of a few elements, how do you model psionic damage, the chilling touch of the undead, etc.? 2) If we're throwing out years of understanding that HPs are an abstraction, then we also need to throw out the abstract nature of attack rolls, redefine the damage expectations of weapons and spells, create separate ways of tracking poison, environmental damage, along with the nature of what type of protection(s) armour provides, how character gain "HPs" (or their equivalent replacement), and how they can regain that resource. That's a lot of work, and I'm not sure the result would be something you could call D&D (unless you worked tirelessly on it for a few years or made the transition to such a game model a progressive process).
I'm pretty sure bards got cures at least in 2e. Paladins and rangers also had healing. The mage/wizard is really the only exception, which is indeed a D&D-ism.
Bards = nope, draws all her spells from the magic-user's (read wizard) list.
Rangers = nope, draws his spells from the Plant and Animal cleric spheres (gets his first spell at level 8, can't get additional, bonus spells, from high wisdom)
Paladins = gets lay on hands at first level, 1/day, 2 HPs per level. That's something I guess (more useful when you need a quick recovery and don't want to waste a cleric or druid's spell). Gets her first spell that can come from the Healing sphere at 9th level (at which time, it's a drop in the bucket depending on how lucky her group got with their hit dice, also doesn't get bonus spells from high wisdom).
Of course not. Who said they did? They do of course have sole control over instantaneous closing of potentially lethal wounds, because that isn't real, but long-term healing, not so much.
But what if it's not a wound? What if it's fatigue, or a psychic attack? What if it's just your "luck" that ran out or your "ability to turn a serious blow into a less serious one" that just failed? Couldn't a not-magical healer do the trick then?
And with better health and healing rules, natural healing and alterations thereof could become more meaningful.
Offering an option or making the default option actually match what HPs represent in the game would be nice, yes.
That's only in your eyes though (perhaps in some others as well). Nonmagical characters aren't mudane at all. As has been well documented, the point at which d20 checks are "realistic" typically stops between 5th and 10th level. High level characters can do incredible things. Skills have open-ended and powerful applications.
Here's my problem, and I'll lay it out clearly. Non-magic man wants to achieve X effect. In first or second edition AD&D, he asks his DM how he can achieve it. DM thinks for a while, assigns an arbitrary difficulty to the task and asks non-magic man's player to roll once or more. The next time that player wants to achieve the same effect, there is no guarantee that the difficulty, number of rolls, etc. won't change.
3rd Edition "fixes" this by codifying a lot of effects into either skills or "combat maneuvres" that a character can attempt. The problem with the latter is that, for some reason, it's punishingly (and arbitrarily) difficult to succeed within the accepted fantasy milieu of the game (I get that it should be difficult to grapple a tank in a modern military game, but making a dragon fall on its scaly ass should be something an experienced character, be she a fighter or a wizard can reasonably do). At the same time, the game segregates the supernatural, mystical, and mythical to the strictly "magical" spectrum of things, while still using mythical, supernatural, and mystical exemplars for classes that have no access to such mechanics.
If you're talking about 17th level characters casting Wish, yes that's something that only wizards/sorcerers can do (and should be). If you're talking about doing useful things in the context of a typical game, fighters remain the most popular class in all versions of D&D (except possibly 4e; that I wouldn't know), and the nonmagical classes in general are more impactful in most situations. Magic offers some fantastical, "limit-breaking" effects, but it hardly renders the other characters irrelevant.
Polymorph, Knock, Tenser's transformation, Divine Power... I could go on naming spells or class features that completely trample the actual abilities of non-magical characters. Knock is really the poster child for this : it's a spell that renders an entire skill pointless. The rogue has to roll a check with a non-impossible chance that he might fail, but the wizard can just wave her fingers, no checks required, and open the door. But, you'll say, she can only do it once, maybe twice, before she runs out. That's when the rogue'll shine! Sure, I answer, but only if there are other locks in the dungeon or if the PCs don't rest before meeting another one. Same goes for fights : in this encounter, the cleric completely outpaced the fighter in efficiency, but surely she'll get a chance to be the hero in the next one.
It's good for a game to make playstyle assumptions, but it doesn't mean you get to not take into account the consequences of the pacing mechanics you included in games with non-standard pacing/games with players who understand the importance of spells/abilities/resources in successful encounters. The best part is, this also works with how HPs relate to the game.
Conan, to name just one. Or even in D&D-inspired fiction : Drizz't, Bruenor, Wulfgar, most of the heroes of the Lance, etc. King Arthur, Beowolf... I could go on.