Inspiring leaders shouldn't exist in fantasy. Got it.
non-magical inspiring leaders - Bard is fine, since it's a full caster.
LOLNope. If it can not shout at someone on 0hp and provide them with enough energy to stand back up it isn't remotely comparable to a 4e warlord. End of story. It can't do the single most important part of the job of a warlord.
Being able to bring allies back into the fight is a critical support function in any version of D&D. Because of how hit points work &c.
A class meant to be capable of support that can't do that is a trap. If the party depends on it, instead of a fully capable class, the party is at a greater risk of outright TPK.
It gets worse, tho....
High level point. Each class in 4e had 10,000 powers (or however many). There’s no way a 5e Warlord can replicate them all.
Pre-E Classes ranged from a low of 90-some powers, (Seeker, IIRC), up to a high of well over 400 (Wizard). Each class had it's own list, by late Essentials, I believe,
one power, Healing Word, had been "recycled" and used by a sub-class of a different class. That is a daunting amount o work for anyone who wants to clone 4e or create new classes for it.
5e provides over 300 spells, and liberally re-cycles them among basically all the classes. Some classes, the Sorcerer the only full caster among them, have no unique spells, at all, while the Wizard had the most at launch, 33.
5e and 4e were very different games, indeed. 5e pointedly cut off design space used by 4e to build powers, particularly martial exploits (which were, overall, narrower in breadth/flexibility than the powers of other sources), so it would be difficult to implement any 4e martial class in 5e, as the Battlemaster illustrates.
2nd high level point, martial classes had more overall power in 4e than they do in 5e. Your not going to see the same strength of effects in 5e.
I suppose that's debatable in terms of very simple, raw single-target damage that D&Ds use of hp makes multiple attacks extremely efficient at delivering. Relative to the hp of their enemies, at 11th a 5e fighter does a lot of damage. Nothing else worth mentioning, but a lot of single-target damage.

As far as versatility and range of contributions, 4e classes certainly beat 5e martial classes cold.
OTOH, 5e casters leave every 4e class in the dust with their sheer versatility/power and glut of resources.
(ie the martial/caster gap was much narrower in 4e)
Unfortunately for anyone trying to add a Warlord to 5e, the support-capable classes are, well, the Paladin (as a support character, most notable for its aura), and multiple full-casters (Bard, Cleric, Druid, the odd sub-class of other full casters). The Paladin is also, in 4e terms, a striker and as close as 5e comes to a defender, in addition to a leader. The full casters are Controllers powerful beyond the dreams of 4e controllers, non-combat problem-solvers, and so forth.
And, we're back in the crab-bucket. A faithful 'port of the 4e Warlord would be a lethal trap for any party that relied on it as their only support character, far too weak and limited to take the place of a Bard, Cleric or Druid. At the same time, it would leave every other martial character in the dust (except, if done right, in terms of DPR). It falls right into the wide & bottomless abyss of the martial/caster gap.