For the record, although I'm in the demographic you're describing I don't think of it as my disbelief being suspended. I think "suspension of disbelief" is in the hands of the individual; the game is already so divorced from reality that to claim "this detail makes it unbelievable" is, imho, B.S..
Rather than suspension of disbelief or breaking of immersion, what bothers me is the crossing of the boundary between that which belongeth to mechanics and that which belongeth to storytelling.
Yeah, this is probably one of the better "anti" warlord arguments.
Conceptually, 4E design had roles and power sources underpinning it. So yes, the Martial Leader looked first at mechanics, whereas 5th looks first to find a class' place in the universe, and builds mechanics to fit that feel.
If you STILL see warlord as a Martial Leader, then you're right, it's a bad fit with 5th.
That said, D&D is still a class-based game on the PC end. And I don't have a problem with a class delivering a particular type of experience. A class, after all, is mostly just a pre-package of "crap you can do." So are subsystems like spell casting.
So to me, it's useful to look back and see what a warlord was capable of when we're considering what it ought to feel like now. But that's not the only consideration. We also have to see whether "Martial Leader" still makes sense. IMO it doesn't. Martial works, Leader doesn't.
So it ends up that a warlord, to me, is a guy that makes the fighting unit better overall. He doesn't have to be "the leader," but he's got to offer some boons and some mitigation. He's not the Fighter, but he's at least as capable as a valor bard, a cleric, etc.
That's the niche I see. Now the mitigation doesn't HAVE TO BE "Inspiring Word." That was traditional, but it could be open to temp hp, reducing incoming damage, target denial, etc.
Worth a thought at least. We don't have to pigeon hole this into Martial Leader Faith Healer.