It doesn't sound like you have actually played a 4e Warlord if you are serious about the "just refluff a bard/cleric as non-magical" suggestion.
Actually, I played a Warlord in 4E quite a bit. But I also recognize that the AEDU power structure gave almost all classes (especially within the same role) the same sorts of mechanical functionality (as I mentioned in my first post.) Cleric grants a divine blessing and gives an ally a bonus to hit the enemy. Warlord shouts a combat tactic and gives an ally the same bonus to hit the enemy. The game mechanic is the same, it's only the italicized fluff that is making them different.
Likewise, in terms of healing... there's virtually no difference mechanically between Inspiring Word and Healing Word. It's merely the fluff that makes one a magical wound-closure ability and the other a rallying "Rub some dirt in it and get back in there!" cry. And this kind of mechanical mirroring occurs across all the classes. A Fighter pushes a foe away a couple squares via a shield bash, a Wizard pushes a foe away via a spell. The miniature's movement on the grid is exactly same, it's just the description of how and why it moved that is different.
We accept in 4E that martial and casting classes have the same power format of at-wills, encounters, and dailies acquired at the same times and in the same quantities (with just the fluff being different)... so why is the idea of a 5E "Warlord" using the same power format of the cleric or bard (IE the spell slot pyramid) harder to accept? I realize that for some people it is... but I'm just not one of those people. I can just handwave the fluff off of a 5E War Cleric and just use the mechanics if I need a Warlord with more options than what I would get with the Battlemaster that badly. But that's just me.
I would venture a guess that WotC feels that an actual class isn't necessary at this point in time (although who knows in another three years if they need something new to put in a book)... and thus they're accepting that those who want more will either scrub the fluff off of a bard or cleric, or they'll create their own sub-classes that will give them what they want (as we can find many already created over in the Houserules forum.)
Personally... I'm fine either way. If they had made a Warlord class in 5E I would have been fine with it, and I'm fine that they didn't. Because anything I don't have that I feel I need I can just adapt or create myself.