Trying to make the warlord fit that role in 5e without resorting to magic or handwaving is going to fail.
"Handwaving" it is then.
On the same token, a fighter archetype is probably too narrow. Ideally, the Warlord should still be a martial class (matching a barbarian, ranger and paladin) but instead of spells, rage, or smites, they get warlordy powers.
There's no martial source in 5e. Instead, such concepts are defined by what they lack: supernatural powers such as spells, spells, spells, granted divine powers, spells, ki, spells, psionics, spells, and, I suppose, spells. (5e has a lotta spellcasters is what I'm subtly alluding to in passing, there, in case anyone missed it.) Probably not coincidentally, the 5 sub-classes in the PH that fall into that all contribute DPR in combat. Two also contribute some enhanced skill use; the other three are decidedly tanky in their DPR contributions. That's not a lot to hang character concepts on. If you do care to use magic, you have quite a range of choices.
That disparity needs to be addressed, and the Warlord, by virtue of having appeared as a full class in a PH1 should be at the front of the queue. It was also the Martial/Leader in 4e, so was in a unique-to-D&D position of enabling relatively normal D&D play without the traditional Band-Aid Cleric, nor any of it's second-string magical replacements (like the Druid, Paladin, Bard, or WoCLW). Suddenly, D&D was almost-seamlessly playable in low-/no-magic campaign modes that had always been problematic, before. Of course, there was more to that (healing surges, marginally consistent encounter design guidelines, formalized Source, etc), but the Warlord was a key part of it.
That way, they are able to do their warlord abilties (heal, buff, tactics), and are decent fighters themselves (when better options aren't available). His role should resemble the paladin's (healer, buffer, warrior, diplomat) but without the divine trappings and powers.
It'd be a nearer miss than the Fighter as a model.
However, there seems to be a lot of "in 4e, a warlord was a nonmagical cleric, so he must be the nonamagical cleric in 5e as well" which to me is both short sighted and ultimately self-defeating.
Martial Leader. The Leader box in 4e was very constraining. They chopped a lot off the cleric to stuff it in there, they were able to split the Druid between the Leader and Controller boxes and still had bits left over - while the Bard dropped right in and still needed some padding. But, formal and narrow though Leader was, it did make a convenient way to reference D&D's long dependency on the Cleric 'type' - the Band-Aid, the healer, the WoCLW with legs - and to easily address that issue. 5e abandoned the term, but not the convenience of having several viable support alternatives. The Cleric, Druid and Bard can all keep a party going when things go south, in slightly different ways, while having a fair amount of versatility, as well. That 'support' type of class is still needed to keep the game running smoothly, but, unlike the narrower leader role in 4e, it's still also tied to magical power.
The Ranger was a striker (like the rogue) in 4e but a warrior in 5e
The ranger still outputs some serious DPR, it hasn't exactly changed roles. Same goes for the Rogue, Barbarian, Warlock and Slayer(Fighter). DPR. I casually juxtaposes with the durability of a 4e defender, in some cases, but without anyting resembling marking. So, not really a shift, more an expansion.
I see no reason why the warlord can't make a similar shift.
Can, and it needs to. A simple translation from the limited source/role framework of 4e would be underpowered.
it "isn't exactly the 4e warlord in 5e" it will continue to be ignored in favor of some mythical nonmagical bard-replacement class. Believe me, the conceptual space gets a lot easier when you stop thinking "cleric-equivalent" and start thinking "paladin equivalent".
The Paladin was a secondary leader in 4e, but primarily a defender, a front-liner. In 5e, defenders aren't really a thing, 'tanks' (I'll call 'em, there's no formal terms) are, they're tough like a 4e defender, and hit like a 4e striker (adjusted for 5e numbers, of course). The fighter, barbarian, pally, they're tanks - even the Ranger presumably could be. Moon Druids, War Cleric, Valor Bards, they're mainly support (and also control, and utility, they're casters - 5e casters are super-versatile), but can off-tank a bit if they had to (OK, the Moon Druid's a bear of a tank at specific levels).
Since the few non-magical sub-classes already available have tanking and skill enhancement sewn up and are all-in with DPR, there's not a lot of point to skewing the Warlord any more in that direction than it already went. OTOH, there's something to be gained in the potential viability of such parties/campaigns in expanding it into the 'controller' space that it also had a clear inclination towards (manipulating enemies, either with clever tactics (INT) or provocation/intimidation/deceit (CHA) which the warlord did in 4e, just only to the degree that wouldn't step on controllers' sensitive toes), as well as making it a viable source of the support a party needs for the dynamics of D&D combat to work.
Of course, I'm looking at it as much from a DM as a player perspective. The low-/no-magic campaign has always been elusive and problematic, requiring all sorts of adjustments, variants, soft-balling, and 'GM force' to get in place & keep rolling. In 4e's brief tenure, it was almost seamless - only a martial controller could have made it better (and I agitated for one of those, too, at the time) - and didn't even have to be a campaign, an all-martial party was perfectly viable in an otherwise normal campaign.
The Warlord - as viable, non-magical support class, any necessary hand-waving included - is not really a lot to ask, but what it could deliver is potentially huge.