The problem is, you can't do that in 5e without magic.
For one thing, there's no Warlord.
Seriously, 5e has neither mechanical deficiencies nor complete conceptual blind spots that prevent it from handling non-magical archetypes from genre or history.
There is a pervasive attitude, born from the inertia of the classic game, and who knows what pathologies from the dark depths of the nerd psyche, that in D&D, magic can do virtually anything, and that, contrarywise, without magic, virtually nothing is possible.
Look at the rules of the early game, before the Thief, the only non-magical endeavor detailed with any meaningful reference to the ever-growing-with-level abilities of characters was combat.
Over 40 years, and the prime contribution of the handful of non-magical sub-classes in the PH is DPR. There's at least a cursory skill system, but in it, skill is just a modest proficiency bonus to a check literally anyone could attempt.
Fortunately, it doesn't end there, there's all sorts of things beyond just big hps and big damage pointing to characters being able to do more than just the most mundane things. They're just scattered about, some feats, a couple fighter features, etc... HD, of course.
Its a matter, on the design side, of bringing them together, scaling them up, and putting together something that delivers.
It's a matter, on our side, of getting other players and DMs to open their minds enough to give it a chance. And, 5e seems like a better environment for that than 4e was.
A Warlord can't remove status ailments like poison or disease.
He shouldn't be able to, literally, he's not some kind of medieval ER doc. But, inspire an ally to ignore the penalties one inflicts for a round or a combat, sure, maybe even make the difference between surviving or succumbing as it runs it's course.
He can't raise a dead ally.
Resurections certainly off the table but saving one at the brink of that last death save should be fine.
He has no access to divination or transportation magic.
Spies and smugglers have gotten information and people and things where they shouldn't be for centuries.
No access to survival magic like goodberry, create food or rope trick.
Amazingly, sufficiently competent and determined people survive incredible hardship without being able to conjure food & shelter from nothing.
And at high level, he's going to lack even reasonable buffs and debuffs that can match mindblank, antipathy, or holy aura.
Antipathy gets subjects to avoid something. Manipulation can do that.
Similarly much of what mindblank does could be mental discipline.
There is no way to match these abilities without magic or extreme handwaving.
You don't have magic, either, without some 'extreme handwaving,' so I'm fine with waving away as much as it takes.
The ally we saw die was an imposter? An evil twin? The artifact hasn't really been taken to the 481st layer of the abyss, it's hidden right in this building? Sure. Call it 'author force' call it melodrama...
And, y'know, that gets closer to a lot of genre bits than D&D magic...
...not that that's a high bar.
There is no way to make the warlord match the cleric or bard in 5e without giving him 9 levels of spells.
Of course there is, just give it equally useful, but non-magical capability. After all, 9th level spells are a 1/day thing restricted to the highest levels.
The best you can do it match a paladin, barbarian or ranger in terms of "warrior with special powers", but a character that tries to do a cleric or bard's job without magic is doomed to fail.
Sorry, no. D&D is not so dysfunctional as that, no matter how many if it's fans may be.