5E warlord doesn't need to do everything a 4E one does
Actually, it needs to do more, or it will be dreadfully inadequate compared to the powered-up 5e versions of it's fellow support classes.
just like a 5E Druid or Wizard or whatever doesn't do everything a 3.5 one does and functions differently than a 4E one.
To be clear: The 5e Wizard or Druid does /far more/ than it's 4e counterparts. In 4e, there was a Druid that was a Leader, the Sentinel, there were also controller druid sub-classes, one that shapechanged (sorta) and one that summoned (in a way tuned to 4e's action economy). The 5e Druid is a full support class, that shapechanges, does area and single-target control, and summons.
Comparing the 5e Cleric or Druid to the 3.x CoDzilla is a smokescreen. Yes, we all know how broken the Tier 1 classes were in 3.5, it was overwhelming. But taking something without that history of extreme brokenness out of the 4e leader box and bringing into 5e as a viable support alternative means powering up, not down, and expanding versatility and flexibility, not paring away most of it.
The question real is, what can we reasonably add to the Warlord to bring it up to snuff? Rituals aren't some separate hand-wavy sub-system for out-of-combat support & utility, virtually unrestricted by class in 5e, they're heavily gated, there's no 'martial practice' type alternatives established. The Warlord /will/ need abilities of the same caliber and level, just, clearly, backed by very different narrative, and working with at least somewhat different mechanisms.
I think the Warlord needs to go further into 'Author Stance' as it gains levels. That would pull the campaign towards a more 'narrativist,' more player-driven mode, so it's just as well the Warlord was put off from the PH and made so profoundly opt-in optional. (Face it, there's optional-in the PH, optional in a supplement, and optional in a late-edition supplement that hits the shelves in the dropping-off fatigued tail of the edition cycle.) But, even so, it's a perfectly legitimate style, and there's no reason not to let DMs who choose to so exercise their Empowerment, opt into things that'd support it.
You can play a low magic game with little or no healing. Just adjust the encounter difficulties to compensate.
You can play a band of bullies wandering around, beating up chumps that have no chance against them. Hps & the restoration thereof is how D&D manages to present the impression of jeopardy, while allowing heroes to survive many 'deadly' dangers over 20 levels. Even then, it tends to have Raise Dead as a back-up. A low-magic game with non-magical healing such as the Warlord might have would still be performing without a net for want of that - a little grittier, a little more meaningful. But, wiith just HD? It's performing without a trapeze, just jumping out there and doing a swan-dive into the ground.
Sure, you can do it from 3' up instead of 30, so you survive every time, but it's not looking so daring anymore, and still no fun...
Edit: OK, that's a bit harsh. There's a level of 'gritty' where that's just the sort of thing you want. You want to face PCs with a stark choice of cynicism - and survival - or heroism, and very probable death. With 'very probable' actually delivering death most of the time.
I'm not saying anyone shouldn't be able to go there - they can, currently, by banning 4 or 5 classes, creating a warlord just gives them a 6th to ban, a trivial hardship, while it opens up the game to less profoundly gritty, potentially heroic, yet still low-/no- magic campaigns.
And, sure, gonzo if you wanna go there...
Right, not to mention that the whole "shouting a hand back on" argument is just dumb and it makes the people using it look silly. It's basically saying "I'm not describing the effects of damage in a way the game intends and, whether I'm willfully ignorant of that fact or just legitimately unaware, I'm still going to base my criticism of non-magical healing on it."
Even Mearls, when he went there, immediately admitted he was being ridiculous.