A full-on 4e Warlord wouldn't fit in 5e's system.
I understand the apprehension, but 5e's system is not that limited nor inadequate. It has set out to cover more styles of play that past editions, not fewer, and certainly not to intentionally exclude 4e and the styles ite enabled.
You certainly can't have it enabling the ton of free attacks in 5e that it did in 4e.
Actually, 5e is quite profligate when it comes to multiple attacks, and has an action economy not dissimilar to that of 4e or 3e.
They just need a more action economy-friendly maneuver than Commander's Strike for the Battle Master, and a better designed leader sub-class for the Fighter than the very flawed PDK.
Neither are really plausible, and get to the same problem that Xeviat alluded to in the OP. Sub-class design doesn't allow for reducing or swapping out base class abilities, and the base fighter class is just too loaded with DPR to have room for cool action-economy-addressing Warlord-style power, nor for the kind of support contributions that the warlord could deliver in 4e. It's not a 5e system problem, it's a 5e fighter class design 'problem' (though only if you want to wedge the warlord in as a sub-class). The solution is a full class that does have design space for abilities appropriate to the concept.
Counter-examples: Warlord and Elementalist may offer novel mechanics, but it doesn't sound like they change the game's story very much.
The Warlord could fundamentally expand the range of campaign/genre 'stories' the game could handle, if it were given at least the range of support contributions it could make in 4e - really, the concept could handle more than that, if it were more fully explored.
The appeal of an elementalist along the lines of the Essentials Elemental Sorcerer is more strictly mechanical, in that it was a simple to build/play caster, allowing players who don't care for mechanical complexity to play something other than a champion for a change.
I suspect I'm opening a can of worms
Oh, the worms are all over already, anyhow.
but I have trouble understanding this desire to largely replicate existing abilities, but "without using magic".
Magic or not-magic is kinda a big issue for a lot of fans. Psionics, in particular, should really be allowed to go either way at the DM's option, as it says something about the setting. Sure, you could re-skin a GOO Warlock as a Psion, the DM could change his Eldritch Blast to a Psionic Blast doing psychic damage, and you could pick mind-affecting and self-only spells and it could work fine. But that doesn't mean the Mystic should be shut down. 5e has many ways to the same concept, even just with the PH.
Another part of it is that we don't really want to replicate the abilities, but rather make the same sort of contributions with a different character concept, and in different ways, with different abilities.
For an example already in the game, look at the Champion, Ranger & Warlock. All three can do a lot of DPR at range. The Champion uses a bow, no magic of any king, the Ranger uses a bow augmented by magic, and the Warlock just blazes away with pure magic. The mechanics are quite distinct, thought the bulk of both the Champion & Warlocks single-target sustained DPR is from making 'at will' attack rolls. Does that mean only one of them should exist? No, of course not.
And 5e is full of examples like that. The idea that some function already being covered obviates any other concept that might fulfill a similar function would reduce the number of classes & sub-classes to a mere handful. It's nonsense in a system like 5e that already gives you so many different ways to, say, do a 'gish' concept - Fighter/Wizard, Fighter/Warlock, Paladin, Ranger, Fighter/Sorcerer, Bladesinger, Valor Bard, Fighter with Magic Initiate, caster with Martial Adept, caster with the Soldier background, etc...
To me it sounds almost exactly like, "I want an archer, with all the same abilities and damage, but I want to use firearms instead of bows and crossbows." I mean, that's great, but it's not D&D, which takes place in a magical (and non-gunpowder) world.
Not every D&D world is gunpowder-free, the 3e DMG had examples of gunpowder weapons, and PF, for instance, /did/ provide a gunslinger class. Gunpowder might seem to clash with the fantasy genre, but, frankly do do a lot of things in D&D - and, a class that provides something other than just at-will DPR as a contribution to the party's success, without using magic, doesn't necessarily class with the broader fantasy genre, anyway.
I'm really just perplexed why there's this desperate intensity to impose this condition on the game when it is almost entirely an interpretation of existing mechanics, not a change or addition.
Why the desperate intensity to resist or deny it? There's already so many instances of several different ways to similar concepts or functions within the existing mechanics, already?
