I love warlords granting temporary hit points, because it works with the conventions and tropes of the role so much better than healing.
You give the big inspirational speech before a battle or rally allies between skirmishes in a prolonged fight. You don't give your big speech to just one person occasionally in the middle of a fight or mostly at the end. But the nature or healing in D&D (4e excluded) focuses on healing after battles.
I believe that, in 3E, in-combat healing was also a significant factor in the game - at least at a number of tables.
There seems to be some disagreement on whether, in 5e, in-combat healing is important. That suggests to me that it is, at least in part, a table thing.
I agree that inspirational healing as a post-fight thing, rather than rousing or rallying, isn't very inspiring. I personally don't like it much for clerics either - when you look at the archetypes for prophets and wandering miracle works, very few of them perform their healing as a form of post-combat medical care.
I reeeeeeeeally don't think we should be using Gandalf as an example of completely mundane inspiration, because he is not a mundane being. Let's also not forget that shortly before this "inspiring and heartening," he had rescued Faramir from the nazgûl with some kind of bolt of white light, in full view of the entire city. I don't think this is completely unconnected to his ability to rally the Gondorian soldiers.
<snip>
I'd be more interested in literary examples of inspiration coming from a completely mundane non-magical human who (1) is not already a leader/commander thanks to some kind of authority structure such as social class or military rank, and (2) does not end up being granted command over others thanks to his/her inspiring abilities (i.e. "We'll all willingly follow you because you inspire us!"). An inspirer among equals, in other words, which is what I think the warlord is theoretically aiming for? (Correct me if I'm wrong about that.)
Aragorn isn't a commander of Legolas or Gimli, but clearly inspires them. (The run across Rohan also shows how inspiration has applications in non-combat contexts.)
There is also a question of how much archetype slippage is tolerable in a class-and-level-based FRPG based around the "adventuring" paradigm. The adventures of most D&D paladins, for instance, hold only a passing resemblance to the deeds of literary knight-errants. And they travel with non-knights rather than on their own, with other knights or with their squires and entourage.
Likewise, as noted above in this post, D&D clerics depart in many practical ways from the deeds of the prophets and miracle workers who (in part) underlie the archetype.
A 1st level cleric can work pretty impressive miracles, and we tolerate the departure from archetype for reasons of playability in a level-based, adventuring-party based, FRPG. I think the same sort of tolerance has to be extended to a 1st level warlord - s/he is inspiring, and recognised as such, even though when looked at from another perspective s/he is not very established as a leader and commander.
Whether archetype examples have to be mundane - I think that's a deeper source of division than the concessions for playability necessary for D&D to be D&D.
In LotR, and other work, Tolkien doesn't have categories of "magic" and "mundane". There is no categorical contrast between the "natural" fear of being attacked by an oliphant and the "supernatural" fear of being terrorised by a Nazgul. Similarly, there is no categorical contrast drawn between being inspired by Merry and Pippin, being inspired by Aragorn, and being inspired by Gandalf.
I think if you reduce all that to
spells, you misfire: spells in D&D are little discrete rituals - like Gandalf igniting pinecones or trying to open the doors of Moria - and in that sense I think it's clear that Gandalf's heartening of the defenders of Gondor does not involve spell-casting.
In mechanical terms it might be an aura or something similar - but I don't think 5e generally forces these effects to be classified, does it (and in that way is more like 4e and less like 3E). I know the Anti-Magic Field spell refers to "spells and other magical effects", but what counts as a "magical effect" is left as a table thing, isn't it? For instance, a table which read the description of barbarian rage ("In battle, you fight with primal ferocity. On your turn, you can enter a rage as a bonus action.") and decided that it was a magical effect that is suppressed in an Anti-Magic Field wouldn't be breaking any rule. And, conversely, a group who decided that a warlord's aura of courage (or however it is handled) was
not the sort of magical effect to be suppressed by an Anti-Magic Field wouldn't be dong anything wrong. Would they?