They also supoprt the model of hp's as wounds just fine.
In fact, the "standard" description of damage includes both wounds and inspiration (though the rules don't take a hard side on it).
The actual hp system allows for a lot of things, yes, it's very, very abstract. Re-imaginings of it, like you have going here, may allow less...
I went into this above, but Second Wind has a few key limitations. The most directly relevant is that an unconscious character can't activate Second Wind.
Obviously, you can't use a bonus action when you're unconscious, yes, and it is meant to be a voluntary choice to call on that extra stamina, not an involuntary reaction.
That has no bearing on the issue your model has with Inspiring Word, however. Second Wind does not require the fighter apply any healing supplies, have a hand free to so much as apply pressure to the wound, make a heal check, nor in any way treat it. Your model maintains that hps cannot be re-gained without either both treatment and time, or supernatural agency. How do you resolve that in the case of Second Wind, which restores hps, includes neither treatment nor time, and is not in any way described as supernatural?
Because, again, "healed enough to not affect you" is not he same as "completely healed."
If you can have /barely healed/ and completely unaffected, it's only a minor matter of degree to get not healed at all and unaffected. You model allows all hps to be restored, without all the wounds that caused those hps to be lost being healed, that opens the door to having hps restored without any wound-healing, at all.
I have to conclude that the Warlord's form of hp restoration /is/ compatible with the very cinematic 'narrative wounds,' model as you've articulated it, to the same level of rigor that HD and Second Wind are. Which is not to say it fits it as neatly as the Healer feat, just that it doesn't demolish the model.
There is healing going on even if the wound hasn't vanished - specifically, some small amount that makes the wound not lethal anymore.
There can be healing going on within moments of a wound being inflicted - clotting, for instance.
A potentially lethal wound does need to be turned into NOT potentially lethal. Doing that without something supernatural requires time (rolling a 20 on a death save is "mystical," for instance)
You can make 3 successful death saves inside of 18 seconds. That's not much time, at all. And, yes, you can roll that 20. There's nothing mystical about it, either, it can function just fine in an anti-magic-shell, for instance, it's a possibility for anyone making death saves, no matter how little mystical ability they may have.
Either your model allows for that, or it is already un-supported by the standard rules, and there's no need (under another of your premises, that, BTW, I also question) for the Warlord to be excluded or conceptually ruined to accommodate it.
If you are willing to arbitrarily make a completely mundane event 'mystical,' in your campaign, to make your model work, than you can do so for the warlord, no matter how clearly it is spelled out as not being supernatural, you can just rule that it is, for your model, to make your model work.