As far as I can tell, the whole point of the Warlord in 4e was that you need a healer, so here's a non-magic class that is a healer. In 5e, I don't want to need a healer, so I really see no need for the Warlord to heal--or even to exist.
Well, healing could just not exist, but that isn't going to reduce the party's NEED for a healer, it will just mean they won't get one. Obviously they'll have to make due. IMHO the most likely result is healing potions. Either that or some relaxation of the lethality of hit point loss, faster gaining, etc. Potions will definitely prevail though IMHO. Honestly, I like them less than clerics.
My real problem is, if you have this morale-boosting inspiration modeled as hit point recovery, then every leader-type has to be modeled as a healer. The hobgoblin warlord has to restore hit points to his soldiers. Genghis Khan has to restore hit points to his archers. Cyclops has to restore hit points to Wolverine. William Wallace has to restore hit points to his warriors. When I think of characters in history and fiction I'd call "warlords," none of them match up with what the Warlord class wants to do.
I don't understand your objection. If hit points are partly morale and inspiration, then how is this an issue. Wallace yells at his men, bucks them up, they regain morale, they can go fight again. Works for me... It seems you're only hung up on this because you're not internalizing the full import of what hit points represent.
When you make something a class, you're not just saying "Players can play as this class." You're also saying "This is a world where this class exists." If you have a class that is defined as "leader guy," that implies that every leader guy has to be a member of that class. It just has weird implications for the world.
NO! In fact this is utterly not true at all. Why would you think that class is an in-game construct? Do you think that people go around in the Forgotten Realms saying "Hey, I'm a 5th level fighter with the Toughness feat!" Of course not. In fact SPECIFICALLY in 4e there are no such thing as NPCs with classes and levels AT ALL. Class is a meta-game construct used to allow players to choose how they're PC will work in the game and regulate how and when it gains features. It has NOTHING to do with the world at all, except inasmuch as PCs are characters in that world that can do things and there are narrative explanations of that, which are likely to be shaped somewhat by the options the players have. CERTAINLY in 4e (and potentially in any other edition of D&D) there simply are no classed NPCs at all and any sort of character is POSSIBLE by the rules. The GM decides which ones exist and what the narrative for them is.
So "William Wallace" could be an NPC in a game. He could have powers of inspiration that are modeled as gaining hit points. He probably will if he's portraying a great battle leader, but so what? He need not have any specific feature of the warlord class unless it makes sense to put that feature on that NPC to represent something you want to exist in the world. It is that simple.
Thus, it would only really work for me if the Warlord were defined as exceptional--not every sergeant or general is going to be a member of the Warlord class. To be so inspiring as to restore hit points requires an uncanny gift that very few have: not just being very charismatic, but having an almost supernatural talent that must be cultivated at the expense of all others. (See the Bard Warlord from that other thread.)
In the end, though, I could just do all that defining myself. So I guess I wouldn't be too upset if the standard game had a Warlord class who could heal. I'd just be grumpy about it.
Well, presumably PCs are pretty special and unusual. In 4e for instance NPCs rarely, if ever, have anything like all the capabilities that a PC has. If there WERE an NPC warlord in 4e its likely to be a leader stat block standard/elite monster with probably a power like Commander's Strike, a single use heal, and some sort of encounter power, probably similar or identical to some warlord encounter or daily. Even elites rarely have more than 4 powers total, and one is likely to be a vanilla melee attack.