Like I've posted before, the Warlord is already in the PHB--Second Paragraph of the Bard description
Nope, Bards are necessarily spellcasters.
Have to imagine that a game with 40+ yrs of history is probably not going to worry themselves with presenting an "accurate" class that has only been canon for for such a short time.
Then you can't imagine 5e, which was conceived as being for fans of all edtions, not just the older ones, and includes newer classes and concepts like the Warlock class, or features like Second Wind or spells like Healing Word.
If the Warlord is supposed to be non-magical, how is it healing someone at range? Morale boost cannot be claimed, because as previously discussed the creature is unconscious.
The whole unconscious = deaf thing has been refuted in the past. C'mon. Alarm clocks. Coma patients.
So, yeah, moral boosts, indeed. 'Healing' is probably a misnomer that should be avoided, though, rather like how Hit Dice are less suggestive of healing than Healing Surges were. Obviously, hp damage can't represent nothing but standardized units of tissue damage, or characters would have to get larger (or denser) as they leveled, and, just as obviously, restoring hps can't mean healing in the sense of making wounds just disappear - that much natural healing can't take place in the mere hour of a short rest, which allows a character who rolls decently on his HD to recover all his lost hps, even from 1. No is the hour required, as Second Wind illustrates. So no-magical hp restoration, be it from morale effects or deep reserves or whatever, is perfectly plausible in the context of 5e D&D.
Like anything else, of course, it can be selectively banned by the DM. Not only needn't any given DM opt-into a hypothetical Warlord class that might give the player the option of restoring hps with a morale effect or the like, but a DM determined to re-jigger the game around meat hps or niche protection for the classic band-aid Cleric could do so, it'd be a simple matter of excising a few inconvenient rules: HD, overnight healing, Second Wind, a feat or two like Inspiring Leader or Healer, etc. That's the strength of 5e, we can all play it the way we like, we don't have to let one tables fun ruin another's just by existing.
I think it should heal, and be named a warlord. The coremost reason of the warlord for existing -from a certain perspective- is to be a nonmagical cleric replacement. It must not use magic, it must replace a cleric and it should be named warlord.
I don't agree that's the primary reason nor all the class should be for, but it's definitely all necessary. The Warlord concept, though could easily reach outside the 4e 'Leader' Role box, and, considering how little conceptual and mechanical ground the non-casting fighter & rogue are up to covering, really should, not just to do the class concept justice, but to explore the design space for non-magic-using PCs that 5e has barely touched so far.
5e doesn't do niche protection so much, so 'healing' in non-divine hands is not an issue. The Bard and Druid are both magical 'replacements' for the traditional Cleric, the Bard having evolved from a Druid-leaning proto-PrC in 1e to an arcane caster in 3e and later, means you can entirely avoid the whole religion angle and still have magical healing available - that's broadened the practically available playstyles the game covers. The Warlord further broadened that in 4e, making no-/low- magic parties, campaigns and settings far more workable than ever before.
5e's goals include both being inclusive of fans of past editions, and supporting /more/ playstyles than past editions - it can't do either of those without filling in the missing pieces that enabled campaigns and styles in past editions. That means psionics and the Warlord, at an absolute minimum, the Artificer for Ebberon, and hopefully more to come.
Maybe they should design the class and then put it with an Aaracockra-like disclaimer. But it oughta heal, and it oughta be non-magical and it oughta be called warlord.
However much reassurance and hand-holding edition-war veterans and traditionalists need to feel re-assured that no one can force a Warlord - or other 4e-isms or non-traditional options - into their campaigns, should be provided, too. 5e is meant to re-unify the fanbase, and if that means take extra care to never breathe the name Warlord without the word 'Optional' in the same sentence (even though everything is optional under DM Empowerment), so be it. The fear of the Warlord is clearly visceral and real for a sub-set of the fanbase, and they need to know that it won't hurt them to give other folks the chance to play the game how they want, as well, and that 'true D&D' - the PH1/Standard/Core game will remain in its traditional form, all sacred cows happily grazing in perpetuity.
Because in 4th ed, the class used a different power source to the one the 5e bard uses.
Technically it used a different power source - a defined keyword in 4e - than the 4e bard used. 5e doesn't have formal keywords like Source. It does still have distinctions like spells being magical, though. Technically, in 5e, it's debatable whether there's a meaningful difference between a Bard, Druid, and Cleric casting Cure Wounds. They're all three casting the exact same, mechanically identical spell, using the slot mechanism. They're all three magical. The distinction that the Bard is casting from a knowledge of secrets passed down in an oral tradition, the Cleric from faith in his deity, and the Druid from a connection with nature is potentially just a philosophical one. The DM might even decide that Clerics and Druids tap the same forces as Bards and Wizards, and merely delude themselves with their religious beliefs - it would make no difference to how any of the mechanics of the game functioned. Yet, while it'd've been mechanically seamless to just have one class taking care of all support casting, and making it skinnable as Cleric, Bard, Druid, Shaman or something else, 5e didn't go that way. It didn't start with the functional contribution of the class, fill in mechanics, and then paint on distinctions between spell, song & prayer. It started with the concepts of the classes in their prior PH1 appearances, and modeled them as best it could. If that led to only one function - like the Champion Fighter's DPR - or many - like the Moon Druid tanking, casting support spells, and casting blasting spells - then so be it.
By the same token, the 5e design philosophy would suggest approaching the past-edition-PH1 concept of the Warlord, an inspiring 'leader' or tactician if we limit ourselves to the PH1 (though other classes 5e were not so limited in fleshing out their sub-classes, so it's not strictly necessary), it certainly suggests inspiration for the former - which could include 'buffs' and hp-restoration, so support orientation - and also some support (buffs) for the latter, as well as possibilities, like manipulating enemies tactically, that would have shaded into 'control' in 4e, but would be fine in 5e.
