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.
I feel that the last claim (on the right-hand side of the 'but") needs more development. Ie,
why is "patching someone up" different from yelling encouraging words?
The game
already allows that someone can push on at full physical performance despite hit point loss, all the way down to 1 hp remaining. So, to the extent that hp model or express in some fashion the occurrence of physical injury or wounding, the game already takes it for granted that such damage doesn't impede performance.
When a warlord "heals" ie restores hp (and I think the technical elucidation of the term "healing" matters here), one need not take it that any physical injury is going away. It is simply continuing not to impede the "healed" person.
I can see the in-principle concern about absurd outcomes eg a warlord keeps "healing" and "healing" so that the physical injury a character is ignoring becomes of Monty Python Black Knight proportions. In 4e this concern was blocked via the Healing Surge mechanic. I'm not sure how one would best handle it in 5e - spending hit dice is one possibility, but (given other features of 5e) seems rather punitive as a mode of healing (whereas in 4e nearly all healing uses surges).
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.
I think limiting to healing half, or the top half, doesn't really meet the concern. It gives rise to a picture of human action in the gameworld where people can push on despite their (lower half) wounds, but inspiration at that point is of no help to them in doing so - but that looks like a rather arbitrary model of inspiration.
Temp hp are an issue for other reasons. First, in general I think they're a clunky mechanic: I dislike the Unearthed Arcana Aid spell, which I think is their first occurrence in AD&D, and I dislike them in 4e, where they cause needless adjudication questions (the answer is normally straightforward enough, but I shouldn't need to ask the question in the first place). But that's probably just idiosyncratic to me. Second, to the extent that they model something, it is being roused before battle "St Crispin's Day"-style; not being encouraged to go on despite the rigours of the actual conflict.
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.
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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.).
Given how many classes in 5e use spells, I don't think there's per se a problem in having another class that double-dips into the Manoeuvre or Inspiration Dice design space.
My dislike of doing it as spells - via bard, paladin, etc - is that it's very hard to get rid of the idea of a spell as a rather discrete magical ritual that draws upon some external power source (eg The Weave, per the sidebar in the Basic PDF).
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
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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.
I've got no strong view either way on what is "too many" classes.
I think my doubts about doing it via Battle Master is a bit more low-level - balance wise, the BM is so invested in doing damage via 4 attacks, using Manoeuvre dice to add damage, etc, that it's not clear that it really has the space to include a rich suite of warlord-y options. I think the paladin is probably a more suitable starting chassis (although it also has some "balance by way of damage dealing" issues), but somehow getting in something manoeuvre-y in lieu of spells (some sort of Eldritch Knight to BM equivalence postulate would have to be the starting point for the design). I don't think the Valour Bard provides a good chassis, because using manoeuvres as a substitute for full casting just seems hopeless.
