FrogReaver
The most respectful and polite poster ever
There is zero reason why a Warlord-like class could not exist within the game. Hell, you can take a Bard right now... make a handful of specific spell selections... remove ALL the fluff from the class, its features, and its spells... and create the foundation of a Warlord. The Bardic Inspiration mechanic stripped of its bardic flavor; Cure Wounds and Healing Word spells stripped of their "magical healing spell" flavor; spells like Bane, Command, Heroism, Aid, Mirror Image and other in-combat spells stripped of all of their 'magic' and 'spell' flavoring and just use the game mechanics by themselves layered with a martial fluff and bent; and if necessary remove the "spell slot" mechanic for determining how often an ability can be used per day and at what power level and replace it with 'martial points' or 'martial dice' to spend instead (or even depower the features such that you can use one every turn at-will if that's preferable.)
Every single mechanical thing in the game can be reduced to basically adding or subtracting a number from another number or changing how often a character is allowed to do something or how they can do it. And how often you can do that, how many characters it affects, and how large those number can get is determined by the level at which the feature comes into play. So just take all the existing mechanics in the game, select the ones that would apply to a 'Warlord' type of class, determine how often and at what character level they can be used (and thus how weak to powerful they can be), and then just fluff all of it with names that denote a military guy giving orders or making tactical recommendations to their fellow characters. You now have a Warlord class for 5E.
And of course the other way to look at it is that the new members of the design team just might not have an issue deciding that for this one specific class that is being introduced 12-14 years into the game's lifespan (at the time of its eventual release) that "hit points are a narrative contrivance" is something they are fine with suggesting this one time. And thus they could be good with just saying that yeah, Warlords can return other character's hit points. It wouldn't be "magical healing", it would be the same sort of morale or energy that Second Wind and the like use as their narrative conceit for it happening. And if a portion of the D&D audience doesn't like that idea? Too freaking bad. They've had 12-14 years of getting their way, so if they don't like this new way of suggesting hit point return... they don't have to use the class.
We are at the point now, more than a decade in, that worrying about "re-establishing the Dungeons & Dragons game" is no longer really on their radar. So they can afford to go further afield with their design choices and aesthetics than they might have done back in 2014. And if that annoys some people? Whelp, people have been annoyed by all kind of things they've done now for the lifespan of 5E, so what's a few more?
Posts like these kind of make me hope they never make a 5e Warlord.