Outside of combat, Help action > the Guidance spell.
"Thanks for the 1d4 when you could've been giving me Advantage."
Outside of combat, you can do both.
I think for me there are two factors involved. First is, simply put, an issue of role-playing. I can role-play a scenario where someone is miraculously healed by magic, because hey... it's magic. It's much more difficult for me to role-play a situation where a non-magic-using Warlord persuades an ally to not be wounded any more. That just doesn't compute for me.
There's plenty of room for restoring hps via inspiration. It's not wounds disappearing. Heck, very little hp damage /can/ be modeling serious wounds, especially in 5e where you have overnight healing. A night's rest won't realistically make wounds, even minor ones, disappear, either, but it does get back all your hps. D&D has rarely had any detail when it comes to wounds. AD&D had an optional negative hp rule that included a minimum 1wk of rest, regardless of hps being magically healed, for instance. Most eds have had a spell like Regenerate to restore lopped-off limbs, even if limb-lopping was restricted to magical effects like a Sword of Sharpness, and never caused by hp loss.
Outside those corner cases, D&D hp loss is very abstract, and, in 5e, with overnight "healing," can't realistically model serious wounds. So there's no reason you can't be at full hps while still bearing some physical injuries - bruises, scratches, stable wounds in the process of healing, and so forth.
I can totally wrap my head around a Warlord ability to let an ally use its hit dice outside of a typical short rest scenario, though. For example, the Warlord might spend an action motivating an ally, at the end of which the ally could spend one of its total hit dice to restore hit points.
That's actually a perfectly good mechanic, and I'm glad you don't object to it. That doesn't make the above objection valid, though.
The second factor is that I think there are enough direct healing classes in the game. I suspect that there will be another one once the Mystic is fully hammered out.
There are more than enough classes that heal with magic, certainly. There are no classes that restore allies' hps in combat without magic, however.
I certainly see no problem with removing some of the magical-healing classes from the game, if you think there are too many. The Ranger and Paladin both cover narrow genre archetypes and could be modeled to some degree with multi-classing and Backgrounds, for instance.
I guess the answer to your question is that I think martial HP restoration should fit within the logic of the game world in a way that is consistent with existing structures. Look at the Fighter's second wind ability
The fighter's Second Wind outright restores hps, in combat, with no action. That seems a pretty adequate precedent. So the Warlord's exhortation is enough to get anyone to 'dig deep' the way normally only fighters can. Second Wind, IIRC, is 1d10 (the figher's HD size) + fighter level. Maybe the warlord could restore target's HD size + Warlord's Level?
I think it shouldn't ever be similar to burning a spell slot to restore XdY hit points.
Spell slots are a pretty specific mechanic, and I wouldn't expect a non-caster to use them. Even psionics aren't quite using spell slots.
For instance, an hypothetical Warlord in-combat hp-restoration ability might be able to affect a given ally once between rests, normally, with any additional uses requiring the ally to succeed on a save and spend a HD. Nothing like slots, but still a limited resource (and it's distinct from slot-based healing, as it'd actually be /better/ for standing up allies in combat at low levels, while not being as good at extending the length of 'day' the party could handle).
When I say "a lot" of hit points, I mean that I'm perfectly ok with the Warlord being able to keep an ally in the fight, stand them up from zero, and so forth. I'm just not able to wrap my head around instantly healing up to full from being critically wounded without using magic or mysticism.
I think the word 'healing' is the main problem. Restoring hps in D&D has generally been called healing, but actually healing an actual wound takes days or weeks or longer, IRL - even a scratch won't heal completely overnight - and, in a medieval setting, is far from certain (you could as easily die of an infection as heal a scratch completely). 5e's overnight healing means that returning to full hit points cannot possibly represent completely healing even minor wounds. There's simply more to hps in 5e than just wounds & the lack thereof.
When it comes to regenerating limbs (just a missing finger, or presumably, other permanent sorts of maiming wounds), sure a mechanic that just restores hps - HD, any possible Warlord mechanic,
Cure Wounds, long rest, etc - simply isn't up to the task.
That's why I like THP as an alternative: it is similar in effect (better, in some situations) and I can role-play the fortification of THP as deriving from the motivation and conviction you get from the Warlord.
THP are a great mechanic, and the Warlord is well-suited in concept to granting those, as well as restoring hps. It should absolutely do both, and do both well.
So perhaps the Warlord should be able to apply a second stat mod (say, Wisdom) to initiative rolls for himself and any allies within 30 feet for him when initiative is rolled. Also, you could give the Warlord an ability (maybe call it "vigilance") that makes any creature making a dexterity (stealth) check opposed by the Warlord's passive Perception to have disadvantage on the roll, and grant that if the Warlord isn't surprised, then no ally within 30 feet that can see and hear the Warlord is surprised, either.I also really like the idea of the Warlord being able to compel a re-roll of initiative once per rest. It doesn't break the game, but it does potentially give the Warlord the ability to turn the tables in an "oh, crap" situation and, as you pointed out, dovetails nicely with the higher-level class abilities that restore resources when initiative is rolled.
All possibilities. The Warlord had a number of features, feats and exploits that affected initiative.
I'm not sold on initiative swapping, though. I don't think having a Warlord in the party should mean that your lumbering, low-dex, spam-in-a-can juggernaut should be able to go first in a fight. Planning for characters that have initiative penalties is (in my opinion) one of the important elements of setting up for combat.
Guileful Switch was about the only exploit that did that, and it was errata'd to make it less abuseable. I'm sure it could be done in a reasonable way, even in 5e's action economy.
One thing arguing against it, though, is that 5e lacks the Delay action (it strikes me as an oversight or over-simplification, but maybe it's for the sake of the DM's convenience & slightly faster combat)...
Edit: or 5e just doesn't call it 'Delay' so no problem, really.
Other ideas: I think the Warlord should be able to use his reaction to negate a critical hit against an ally he can see within 30 feet, once per rest.
That's a fine idea. It's unfortunate that 5e dropped the 'interrupt' terminology, since it raises the potential to mis-understand 'Reactions' as actions that must happen after the trigger is resolved, which is clearly not the case - this kind of ability could run afoul of that misunderstanding.
Also, because it is important for allies to remain relatively close to the Warlord, anyone within his "coach bubble" should maybe get advantage on saving throws against area of effect attacks.
That's getting back to the Commanding Presence, close burst range of inspiring word, and wargame 'command radius' - and, yes, it makes some sense. It'd be simpler to have one such rather than have each ability operate differently - it would, of course, be more flavorful and potentially address some realism concerns to have different abilities function differently.