I haven't chosen to reply to this particular post out of any intention to single it out. It was just an easily-quoted, clear statement of the point.
No worries. I take it as a compliment that my argument is coherent, whether or not you agree with it. Now, I'm obligated to respond, though.
My response: I find the idea a bit odd that inspiring people, and thereby influencing the extent to which (i) they try as hard as they can, and (ii) they succeed when they try, should be cast as magical. (Unless we mean the everyday "magic" of human emotional responses - which are often a focus, and even a somewhat exagerated focus, of dramatic and melodramatic narratives.)
I think it has a lot to do with the abstraction of hit points, specifically, though there are probably other factors. If you view hit points as being even somewhat physical damage, there really isn't a good way to have complete healing that isn't magical. You might be able to say that someone uses a healing kit to patch them up enough that the wounds aren't impairing, but that's a lot different from yelling encouraging words from across the battlefield. Even if you take an approach where only the last, say 10 hit points (or bloodied, Con score, etc.) are actually physical damage, there's still a lower bound where it just doesn't make sense for the commander to be able to motivate the troops "better".
This isn't to say that I don't like the idea of non-magical healing. Quite the contrary. I actually really hate playing Clerics and, as a DM, typically find them the least compelling characters in any group -- I usually encourage Bards over Clerics for this reason, but no one in my group likes Bards, either. I just don't care for martial healing that is functionally identical to magical healing. If the Warlord granted temporary hit points or could only heal up to half (or only the top half, though that sounds punitive) or had some other something, that would be a bit better.
In 4E, I kind of gave the Warlord a pass on this because the whole system seemed to really embrace the abstraction. Ultimately, that's probably one of the big reasons why my group stopped playing 4E: the abstractions were so great that it stopped feeling narrative and started feeling more like a board game with rules that were thematically and balance appropriate, but required too much suspension of disbelief for an RPG (please no edition wars, YMMV). There were a number of character (and monster) abilities that were theoretically mundane, but really smelled like magic because of the way the rules were built. If you were down with Wuxia or Diablo, it worked out, but it wasn't as tight with a LotR or Conan tone.
In 5E, if you're OK with a Warlord who has that pseudo-magical ability to heal and motivate others, then I don't see why a Valor Bard wouldn't work. So what if a lot of the abilities are called "spells"? 5E has (rightly, IMO) gotten rid of the distinctions between divine magic, arcane magic, etc. It's up to the group to decide whether the Bard or Ranger are tapping divine sources, calling on nature spirits, or got their abilities through study. I would have liked to see them step back a bit from the "mystic sounds" of the verbal components, though. Maybe your Bard only does magic in the sense that he's able to tap into the deep subconscious of his companions. The original concept for the Paladin wasn't actually a holy knight -- that was the Cleric -- it was someone who was such a shining example of nobility (in the moral sense) that he was blessed by the powers that be. Maybe the Warlord could be seen as someone who has such a powerful aura of leadership that even the threads of reality bend for him. The Valor Bard certainly works for that.
If you want truly and explicitly non-magical healing, and are not happy with existing constructs (not saying you should be), then I think it would make sense to either expand additional options or create a whole new framework. The former could include some new Maneuvers for the Battle Master that added healing, etc. This would be my preferred route, as I generally find the Battle Master to be a bit lacking, anyway, but like the core concept. Adding a new class and framework could work, too, but seems like extreme overkill because you'd either end up with something that was just a spell list by another name or otherwise redundant with something else existing (Maneuvers, feats, etc.).
I guess I just don't see the case for a completely new class. I can see the argument that the Battle Master should be given the Ranger treatment, and I wouldn't be opposed to that -- I'm just not invested in it. I oppose a new class because I think there's an upper limit on the number of top-level classes the game can handle before the system shifts from "accessible but rich in capability" to "complex and full featured, but not accessible". D&D works best when it's accessible with support for moderate complexity.