All of these problems stem from the idea that real physical wounds "can't" be healing that quickly, or that only certain fantastic power effects in the game are "magical enough" to "really heal" an injury.
Why not do away with this prejudice, and watch the system work perfectly? Take the word FANTASY in "fantasy role-playing" to mean, well, FANTASY. Take the words POWER SOURCE in "Martial power source" to mean, well, POWER SOURCE.
Real wounds simply "can't" close that fast, and especially not in response to some Warlord yelling some pep talk? Probably true enough, in our real, non-fantastical, non-heroic, non-adventure, non-fictional world, the one in which we live and which our experience tells us is governed by certain nearly-indisputable scientific realities.
I'm confused, though, as to which part of the D&D published material led people to believe that this game is set in such a world. As far as I can tell, the assumed setting for D&D is vastly more magical, mystical, wondrous, unusual, epic, larger-than-life, and divorced from our typical expectations of the limitations of "reality" than that.
I'm confused as to which part of the D&D published material led people to believe that the protagonists in this world are essentially just like us normal folks here on mundane Earth, save for having a few neat tricks up their sleeves. Apparently, from my reading, the characters are HEROES, they're the best and the brightest, the ones who can do the things that other people can't do . . . and this, in a dramatically fantastical world!
Why can't everything they do be "magical" in some sense? Maybe not in the big, flashy, drawing-upon-external-forces-beyond-this-world sense, like Arcane and Divine magic. But a more subtle magic, a more personal, inner magic, that comes from their own heroic core. Why can't the Martial "power source" be, you know, a POWER SOURCE? Why do people almost universally insist that you've got Reality-Defying Wielders of the Arcane Power Source of Mighty Arcane MAGIC, and Reality-Defying Wielders of the Divine Power Source of Mighty Divine MAGIC . . . and oh yeah, some other guys who are basically just normal shmucks that happen to be decent with a weapon and foolish enough to brave the unknown.
Clerics and Wizards apparently get to partake in the magical, mystical, fantastical, epic, heroic, world-of-wonder awesomeness of the game and setting, they get to be and do things that no one in our world could ever hope to do. But Fighters and Warlords? Nah, they're not magical in any sense, they're basically no different than a regular skilled martial artist or athlete here on Earth!
Bollocks, I say.
I say that the Martial power source is a "power source" indeed, not merely the retarded cousin left over after Arcane and Divine got done dividing up the fantasy coolness pot between them. I say that it's not "unrealistic" for a HERO in a FANTASY world to recover from grievous wounds in "miraculous" time. Know why we think it's miraculous? Because we're judging from the perspective of our own non-heroic, non-fantastical lives in our own very, very mundane world!
The idea of some guy throwing fifteen-foot radius bursts of magical fire around all day long, out of the thin air, without ever breaking a sweat . . . that we can totally buy, because it's "fantasy" and "magic". But a hero on par with that level of power who . . . heals . . . more quickly than we do is apparently preposterous!
I think people have this idea that since Arcane power and Divine power are the big, overt, externally-derived sources of dramatic effects which are most often associated with "magic" that it must mean that anything else, no matter how heroic or how fantasy-based it is, must be NOT-MAGIC. At all. So no fast healing! I want Wizards and Clerics and real-life Jackie Chans only. If we can't do it in real life, I don't want our so-called heroes doing it in-game . . . unless it's a Fireball. If it didn't come out of a spellbook or a holy symbol, I'll be damned if anyone's gonna do anything "unrealistic" or "miraculous" in MY super-heroic epic fantasy game!
The game doesn't say that Warlord healing is somehow "less real" healing than Cleric healing. Know why? Because it isn't. Yes, a person here on Earth probably can't make wounds close by shouting encouragement. A person here on Earth can't fight liches and dragons with a hand axe, either. The Warlord isn't some Marine Corps sergeant who's listened to a few Anthony Robbins tapes and memorized a speech or two from Shakespeare's history plays. He's a FANTASY HERO, using not merely the mundane resources of the common folk, not merely the absence of Arcane or Divine power, but the equally potent and fantastical Martial POWER SOURCE. It may not be "magic" in some limited sense(s) of the word, but there's no reason to assume that it isn't magical in terms of how it can affect the world.
There is no disconnect, no Schrodinger's Wounding, no need to make up narrative contrivances either before or after the fact. Not if you simply take the game at face value. Fantasy world, heroes, these effects all "heal" in the same way, wherever the power came from. Done.
I daresay that if you deny this simple principle, you're making the cover of the PHB into a liar. According to the standard paradigm I've seen espoused in these threads, it SHOULD read: "Arcane and Divine Heroes, and also, Normal Folks Who Do Some Martial Stuff".
Yes, that Inspiring Word DID just cause my severed femoral artery to spontaneously fuse itself back together. Cool, huh? I guess that's why they call it a "power source", and why they call him a "hero".