So we're getting to the core of this. You want non-magical spike healing, period.
I want non-magical spike healing
if the Cleric's main healing is going to follow that approach. Which is why the OP started off by being about clerics. If the notion of healing in D&D is going to be fundamentally changed from any previous edition to the point that it is removed from the cleric then all bets are off. (If Cure Wounds starts working the way Lesser Vigor used to (1hp/round for a set number of rounds), we're playing a different enough game that my comments are irrelevant).
If spike healing exists other than as a consequence of seriously high level magic (I'll drop the spike healing requirement if the only spike healing the cleric gets is
Heal which they can't get for a lot of levels) then it is by far the most tactically important kind of healing, and for many reasons much more effective per hit point than pre-emptive damage mitigation.
My entire point in the OP was that the warlord must be able to trample on the cleric's ability to keep the party on thier feet and contributing usefully to combat - they must be able to take on a measure of this aspect of the cleric in games both with divine magic and without. This means (amongst other things) keeping them on their feet, and this means spike healing. Alternative models of healing are much,
much less efficient at this than spike healing - and much less evocative.
So if spike combat healing is the assumed baseline (as defined by the way the Cleric does things) then yes I do want it, period. If it isn't then the stark loss of comparable efficiency goes away.
Which means that all this barking around the idea of genre and commanders is irrelevant.
The genre and commanders are absolutely relevant especially as "
hit points are D&D's mechanic for genre emulation". If you want to redefine D&D so that hit points change from the primary (and normally only) form of attrition that is not entirely PC controlled then the context I am talking in changes. If we were playing a game in which the PCs tracked morale points as well as hit points and were more likely to break and run than actually die, things would be different.
If in the Ravenloft module the Warlord primarily prevents fear that works in Ravenloft, which has a lot of fear mechanics and in which fear is an important part of the game. And which is a much more unpleasant setting (from a PC's perspective) than the D&D baseline.
The thing is, non-magical spike healing doesn't need to be in a unique class.
The thing is a class is the single easiest thing to add to or remove from the game. Indeed I believe that every class other than the fighter and the rogue should have a paragraph discussing what its presence or absence means to the gameworld and the impact on the metagame - and on a party. For instance a game with (using 3.X examples) only wizarding arcane magic, with only sorcerous arcane magic, and with only bardic arcane magic are different worlds.
"Doesn't need to be" is not the same thing as "There is a good reason why it shouldn't".
A "heroic rapid healing" module could easily contain, heck, the exact mechanics of Inspiring Word, without a warlord class.
And the problem with this is....?
Once again you are fundamentally redefining the relationship of the game both as a game and the game with the fiction. If everyone gets Inspiring Word then we've almost ended the concept of focus fire and reduced the tactical considerations to "Squash the squishy and finish them in one turn or they'll just stand up again."
And inspiring needn't be the same as healing if you have a richer hit point model, but needs to restore hp. In 4e they were mechanically distinct (although clerics could do both) because inspiring involved allowing someone to spend their own resources (the healing surge). Only rare magic healed with no healing surge.