5e classes single attack are generally weaker then their at-will (multi-attack) and short rest (action surge) or daily (spells) abilities too. At least past level 5.
Except the rogue.
Yep, 5e's design philosophy requires a little more thought & detail when it comes to how class features interact. One of the more practical reasons for not adding classes. Any new class that's not very basic in abilities (not just another DPR-focused class, for instance) is going to run up against that difficulty in a way that's a lot more obvious, but no more serious, than it was for classes that were developed or or less simultaneously and behind the scenes.
I thought I had.

. Most of the elements that made the 4e Warlords niche have either been passed off to other classes or don't really need to be in 5e.
Even if both of those were true, they wouldn't mean there's no 'need' for the Warlord in the game, just that there's no absolute need for one with in a given party (which is fine, and how it should be for every class: choice is a good thing).
First of all, the Warlord didn't have a protected 'niche' the way an old-school Cleric was a band-aid or a 3.0 Thief was the only class with the Trapfinding feature. It just had a formal Role, one that it was never alone in being able to fill.
Secondly, spreading a concept over several classes, while also having a single class or sub-class that can more neatly and fully cover that concept, is SOP for 5e. You can play a Fighter/Cleric or a Fighter with the Acolyte background - but you can also cut to the chase and play a Paladin. You can play any class with Magic Adept and a background like Sage, or an Eldtrich Knight or Arcane Trickster, but you can also play an actual Wizard. Being able to half-posteriorly cobble together a baroque build that suggests a concept doesn't mean there's no need for a class that embodies it.
Tactical movement forex. In 4e, that's a major element.
That's a little overblown, IMHO. 4e had a pendulum-swing away from the 'static combat' complaints of 3.x, and 5e swung back the other way. No bearing on whether a class concept is 'needed.' Every class in 4e had some tactical movement built into some powers and other powers that had nothing to do with tactical movement. The Warlord was no exception. Those sorts of abilities are handled differently in 5e than in 4e, not done away with.
You couldn't move-attack-move in 4e. In 5e it's pretty easy to be highly mobile on the battlefield without any help.
As it was in 4e. Every class had some tactical movement stuff, just like everyone could make a basic attack. The Warlord might grant tactical movement or basic attacks as part of his build, or not. The whole class didn't rest on either of those things.
4e had a bajillion status effects. Having a warlord that could mitigate or remove them was a big deal.
Fewer than 3.5 but more than 5e - though, really, not a lot more standard ones. The ones 5e does have can be pretty brutal, and even if they might let you save every round, a bad save can be pretty hard to make.
In combat healing is largely not required unless someone drops.
As has generally been the case since 2e. In 1e, you had to avoid letting your allies get to 0 hps because they either died or (optional 'death door' rule) would need a full week of rest to recover, no matter how much you healed them. Once that rule was dropped, the tactic of healing only dropped allies became viable. In 4e & 5e, which both used heal-from-0, it became optimal.
It's much more like Adnd where healing is done afterwards. And since everyone can burn HD on a short, you don't really even need a dedicated healer in the group. Paladin or Ranger can cover a lot of that.
Obviously, Healing Surges available after a 5 minute rest were a /lot/ more available in 4e than HD, useable after an hour-long short rest, even if they didn't also represent a far greater hp resource. Similarly, 4e and 5e both allow in-combat healing to be done while still taking a normal action, like attacking. The only way in which 5e tends to have less need for in-combat healing is to the extent that it's combats are so much quicker and easier. If you run a lot of (or even any, really) hard-deadly combats that aren't just a quick mopping-up exercise with 0 risk (either intentionally or because the encounter guidelines fail you), you'll be needing a party that can stand up dropped characters.
So what's left? If you don't need tactical movement, and you don't need in combat healing, and mitigation of status effects aren't as important, what's left?
I suppose 'need' is relative. You don't 'need' in-combat healing, but the Cleric, Bard, Druid, Paladin & Ranger all have it via spells, including Healing Word that can be used as a bonus action while attacking. You don't 'need' tactical movement, but the Rogue still gets Cunning Action, the Wizard adds Misty Step, races still get different speeds, &c. You don't 'need' to mitigate status effects but the Cleric, Bard, Druid, Paladin & Ranger still do it. So either you do need them, or 'not needing' something is no reason at all not to have it.
In some ways, 5e's less solid/more open handling of those (and other) things makes the need for a class that can model the concepts the warlord covers that much more acute. In 4e, you could play a 'tactical' character even if there were no Warlord class, because the detailed handling of a few aspects of tactics, like movement & positioning, let the player step in and provide his own, meaningful, if technically metagame, tactics that looked in-fiction enough to be satisfying. 5e doesn't currently support a that style in the narrative very well (even with the tactical module), and metagame tactics revolve around spellcasting. A warlord, at the very minimum, could add some tactical depth via more than just spellcasting choices. It could also model some of what 4e did at the gamist level of player tactical decisions at the narrative level with a more abstract modeling of the character's tactical ability.
For instance, in 4e, Tactical Presence had a mechanical effect whether the player (of any class, not just the Warlord) was applying any of his gamist 'tactical' (play) skill well (or at all), but when it came to tactical movement, the player had to make a good tactical choice about positioning. In 5e, tactical positioning could be handled more abstractly, the player could use a maneuver that just "moves an ally into a tactically advantageous position" without having to come up with such a position, himself, based on the mechanics.
I rather hope, actually, that a 5e warlord is based on the flavour- tactical/inspiring leader, but I think mechanically, you can't bring much of a 4e warlord forward into 5e.
You can't, now, you could very easily. The mechanics needn't even be all that different: most of the classes in 5e were designed with mechanics that called back earlier editions. There's only one earlier edition of the Warlord.
I'm just not sure what's needed to express the concept in 5e.
A martial class that fill the party's primary-support needs and models at least the same range of concepts the 4e Warlord did. That requires mechanics - probably a maneuver system and/or a completely novel system tightly coupled to inspiration/tactics/etc - that are customizable & flexible to a much greater degree than those of the few classes (Fighter, Rogue, Barbarian) that have non-magical sub-classes in 5e. You could just port the 4e Warlord into 5e. It wouldn't
need to be as wildly versatile as 5e casters (no class does).
A good analogy might be that the Warlord needs to be to the Battlemaster what the Arcane Trickster is to the Wizard. Less (more situational) DPR, vastly more choices/resources.
Well, I'm about to test that out once my Primeval Thule campaign gets off the ground. About the only healing, outside of potions, might be a ranger since I'm not allowing any classes/races with at-will spells.
The Ranger is about the worst support-caster in 5e (fewer support spells and far fewer slots than the Druid, Bard, or Cleric, and no LoH like the Paladin), but it's still a caster, so hardly 'all-martial.'
Unless healing-potions are re-skinned as non-magical, they're pushing it, too - and if they're acceptable as non-magical healing, then a major spurious objection to the warlord restoring hps is gone, as well.
I know our current Dragonlance campaign, which has only a paladin and a ranger (out of 6 PC's) for healing has been doing fine and dandy up to 9th level now. So, I'm not really convinced that an all-martial party isn't viable.
Neither the Paladin nor Ranger are all-martial, and the Paladin is actually quite a capable healer, by itself, thanks to LoH on top of cleric-lite spells.