You want a nonmagical warlord to have nonmagical solutions for... lost limbs, lycanthropy, and petrification.
Weird, it's almost like you picked a bunch of obscure, rare status effects that are explicitly magical in nature to prop up your argument, instead of the much more common stun/charm/frightened conditions that a warlord should be able to address. Funny, that.
I have no problem with the warlord not being able to address every condition under the sun, but the usual ones you'll commonly encounter should definitely be accounted for.
Funny thing: I've had people argue just as strenuously as you are now that short-rest healing is inherently overpowered.
It might be, if the example amount was of any significance.
You talk as though the class you apparently want the warlord to be -- the cleric, or perhaps the paladin -- were not gated with *gasp* long rest resources.
Maybe you should stick to arguments presented instead of postulating as to other poster's desires. First off, even if that was the case, there would be nothing wrong with that mechanically. We have a short rest arcane caster (the warlock), and it works just fine, and even the wizard gets spells back on short rest, so it's not like there isn't already precedent for that design paradigm. Secondly, no, what I want is a distinct class that can fulfill the same role as a leader archetype to the extent that they could be a viable replacement for the classes that are currently assumed to fill that role, i.e. cleric/bard/druid.
I have no issue with a magical subclass of the warlord being able to do all that, but in my expectation of the non-magical warlord it will not have tools to deal with lost limbs, lycanthropy, curses or petrification. I could see them having tools for helping with poison and (to a lesser extent) disease, but those would not be the immediate curing kind of tools that magical characters like clerics would have. I can also see a warlord having tools to help with certain status effects and with drained stats.
And that's all perfectly fine, for one very good reason: the warlord is a combat-focused support character. They should be able to heal and help with some of the things you'd generally want a cleric for, but that kind of support should have a built-in cut-off for kinds of things it can't do. That's part of what makes it okay for the warlord to perform other support actions, like enabling off-turn actions, without doing too much.
Agreed, and as above you'll see in my response to cosmic, the more esoteric ones are fine to be out of reach for the warlord, but that's not a valid reason it shouldn't be able to counter charm/fear/stun effects and the like.
The point is that the war cleric isn't a healer. The trickster cleric isn't a healer. The tempest cleric isn't a healer.
The cleric isn't always a healer. It can be the tank, it can be the face, it can be the sneak, or it could be the blaster.
So why does the walord have to be the healer when even the cleric—the class synonymous with the healer role—might not serve as the healer? Shouldn't that be more of an option in a subclass of the warlord? Why is healing so essential to the class?
A trickster cleric can heal. A war cleric can heal. A tempest cleric can heal. It's the unifying baseline of the class, much like all rogues sneak attack. You're free to play a rogue that never sneak attacks, a mage that never casts spells, and any other odd-ball option you want, but to pretend that this is the common approach is farcical to me. I've certainly seen clerics built in a large number of ways but I have yet to ever encounter a cleric that never preps healing spells to some degree.
How exactly does a warlord bring someone back from the dead? Back from stone?
It doesn't, nor should it.
As I say above, in the 5e design a big role of the "healer" character is casting spells like lesser restoration or raise dead. Which the warlord cannot do without actually casting spells or having magical abilities. Thus it cannot fill the role.
It can fulfill the role, just not to the same extent, in that the warlord is worse at reacting to effects already placed on PCs and better about proactively dealing with them. I would liken it to the fighter, barbarian, and paladin. All of them can fulfill the role of melee front-line combatant, but they do so in different ways and with some options not available to others.
Meanwhile, classes have a finite number of abilities. 16 or so class features spread over 20 levels.
Each time you add a healing one to the warlord, it takes away a warlordy feature from the class. Something unique to being a tactical leader and commander that no other class could do.
Especially at low levels. Because, if the warlord is the healer and all warlords despite build have to heal, then that's their first level feature. It's making restoring hp a more iconic part of the classs than granting actions or movement or increasing initiative or buffing ally attacks.
The design of the warlord should focus on the cool things of the class, not the expected things of its 4e role.
I would totally cede the healing point if people would stop doing several things:
1) trying to staple the class onto the fighter. It'll never work as long as it has that baggage and misspent design space and it's exhausting to see this repeated over and over and people still not get it.
2) being incredibly gun-shy about action-granting/enabling. Even in the OP, cosmic later clarified that granting cantrip use with the class features was too much, and this mentality needs to be dumpstered if a 'real' warlord is to be made, because sans the healing option being able to grant actions will be the defining aspect of the class, and thus necessary to be used in as many situations as possible.