It wrecks the narrative. That's all it needs to do.
I see some pretty easy ways to work within the 'narrative-wound' model to make Inspiring Word and other such abilities work pretty well, indeed. As a matter of fact, I'm starting to get just a bit enthused about it.
You can't ignore wounds forever.
You don't have to. Assume they're tended to mundanely at the earliest opportunity. Don't even have to do it 'on screen,' very much in keeping with those tropes you posted up thread.
That's how inspirational healing fails - it is not consistent with the wound narrative.
I don't see why the narrative would have to fail to account for inspirational healing: You get wounded. You rest an hour. Your wounds haven't healed, but you're still at full hps. You get wounded. You reach into your deep well of stamina and fight on. Your wounds aren't healed, but you've re-gained some hps. You get wounded. Your friend reminds you what you're fighting for, and you fight on. Your wounds aren't healed but you've regained some hps.
And, if you want to add 'you bind your wounds and..' before 'rest an hour' and '...until you have a chance to bind your wounds' after the other two, for a nod to reality, you certainly could without touching any mechanics.
Much like gods, Lovecraftian entities are a presumed part of the game narrative.
Hypothetically, if they weren't, would that make adding the GOO Warlock objectionable? By the logic you're using,
yes it would. Which, sadly, seems to boil down to 'nothing new can ever be added to the game - new takes on existing things, at most, are allowed.' That's just not helpful.
Beside, just because the game has a narrative element, doesn't mean everyone at the table wants to deal with it. The player who brings in a GOO Warlock brings the Lovecraftian element to the party, 24/7, for the whole campaign.
Accepting and even appreciating or at least working with other players' choices is just part of the game. A necessary part. Restricting player choices is also part of the game - part of the DM's role, when he wants to restrict or establish something about the world. If a DM decides he's going to have a less creepy setting, he can toss out the Far Realm and disturbing Lovecraftian monsters - and the GOO Warlock as a player option. If he decides he wants a campaign where everyone graduated from the same Wizard School, he can allow only Bard, Eldritch Knight, Arcane Trickster, Sorcerer, Warlock or Wizard as player choices.
Entirely inspirational hit points are not a presumed part of the game narrative, though it's not incompatible with the rules as they stand. Entirely wound-based hit points are also not part of that presumed game narrative, though they are not incompatible with the rules as they stand.
And the two turn out to be compatible, in a narrative-wound model like the one you've articulated!
You can accept that, to handle Second Wind in the narrative model, hps can be restored, without any healing or treating of the corresponding wound, and without magic. Which have the added bonus of working to make Inspiring Word acceptable, as well.
You can imagine, as you did with Death Saves, that there is some subtle, semi-mystical force at work (that 'well of stamina' is not perfectly mundane). And, as an added bonus, you can assume the same for Inspiring Word, even if, as with Second Wind, there is nothing at all in it's mechanics or fluff nor that of the class, to imply that it's 'magic' or supernatural.
Any reason either of those couldn't work for your model?
That's a campaign-wide decision, though, not a decision that is in the hands of one player making a class selection.
The player making the class selection wouldn't be making the decision. The decision isn't whether to play a Fighter or Warlord, but how the Second Wind and hypothetical Healing Word abilities would be handled in the narrative.
In the first case, the DM or DM & table realize that the time & tending wounds or magic requirement can't apply to Second Wind, and decide to drop that requirement - or moderate it by letting the wound-tending happen later.
In the second case, the decision can be made independently, since it's just how the individual player (or PC) believes the ability is working. Even as a player, you can decide that a Death Save or Inspirational Healing 'must have been' the work of some semi-mystical agency, something too subtle to be detected or stopped by an anti-magic shell, but more than mundane, none-the-less. Not provable, but you could have endless, pointless, debates about it, and no one's narrative is 'wrecked.'
So, any reason either of those couldn't work for your model?
(Of course, there's also the DM option of house-ruling second wind and death saves to make them conform more closely to the narrative model. But, at that point, preserving the model is not longer an issue to consider when designing the Warlord, the Warlord could be added to the Standard Game, and, like Second Wind, could simply be banned our house-ruled.)
Yeah, if that was available within the presented narrative in 5e, that'd work fine.
In general, 5e presents non-magical, non-supernatural powers in a fairly straight-forward way. It doesn't say they're magical or supernatural. OTOH, Ki or spells, it comes out and says 'look this is magical.' Fans of the Warlord (and it's detractors) may be adamant about the Warlord being non-magical, but all that really requires is the lack of any rule saying or implying that it /is/ magical. Even if the fluff text firmly states that the Warlord is strictly mundane (which'd be pushing it, there's nothing mundane about heroism in the fantasy genre), it's just fluff text, and an underlying mystical rationale (one not mechanically interacting with rules that apply explicitly to magic or other supernatural powers) could effortlessly be assumed, if that helps with visualizing the power. (Hopefully not to 'shout wounds closed,' but to provide a preternatural fig-leaf to ignoring wounds or fighting on for those who just aren't quite satisfied with 'ordinary' heroic fantasy characters doing so unaided.)
At this point, not only do I not see how a compromise is 'impossible,' as you asserted up-thread, because of the hp-modeling issue, I see no related reason the Warlord shouldn't be added to the Standard Game. Those who are already using modules and changing rules to enable a hp model can simply ban or modify the class, since they've already moved beyond the Standard Game, anyway.