The problem here is that 4e's warlord is a child of 4e's design paradigm. Specifically, what a "leader" can do.
That's not a problem, it's an opportunity (did I really just say that?). What a leader could do in 4e is /less/ than what the same class can do in 5e. The Warlord wasn't defined by the Leader role in 4e,
it was limited by it. 5e removes that limitation.
Take a 4e cleric vs a 5e cleric. Think of what the cleric's powers in 4e do. Mostly, they either a.) heal; b.) buff, or c.) do radiant damage*. Most of the "support" spells were rituals anyone could learn (such are remove affliction, raise dead, or divination).
Rituals were a decidedly minor part of 4e play, but FWIW, 4e also introduced "martial practices," so you could avail yourself of a similar system without resorting to 'magic.'
I wouldn't worry too much about the way they've been folded back into spellcasting (an actually fairly elegant way, really, compared to being tacked-on and draining gold).
Take away the rituals, and a cleric has far more limited range of effects, which can be either replicated (in a martial form) or swapped for similar effects. A warlord works as a cleric replacement because the cleric class had a fairly narrow range of effects.
The Cleric had quite the breadth of powers, with both weapon & implement powers, numerous ranged, area, and radiant powers, a variety of utilities, etc. The Warlord did not simply replicate them, it had quite different martial exploits. The two classes were the same role, and both reasonably balanced in that role, but they were very different.
The premise that they have to be the same to fill in for eachother doesn't seem to wash.
In 5e, it's even less clear that you have to do all the same things to step in as the 'support class.' The Cleric, Druid and Bard all do support stuff, but the rest of what they do is somewhat different - in spite of the profound overlapping of spell lists - the Cleric might turn undead, the Druid turn into a bird and spy out an area, the Bard turn up bits of vanished lore. A given Cleric, even, might be pretty inept in battle or a near rival to the fighter through mid levels.
One of the great things about 5e is that it does not, as 4e did, put classes in boxes. In designing a class, it's possible to overlap with a concept or function of another class without obviating it, because there are also going to be areas that don't overlap and grant conceptual and mechanical distinctiveness.
I'm not saying your concerns are unfounded - from the 4e paradigm, it's hard to look at 5e as anything but radically imbalanced and 'incoherent' (much as I dislike the Forge term) - and thinking in terms of that tighter, more balance(consistency/clarity/etc)-oriented paradigm, when it happens to be the one that brought us the Warlord is perfectly natural.
But the 5e paradigm embraces /more/ than the 4e paradigm, not less.
That's a major point of it, and how it reaches across edition boundaries to all fans of D&D.
(....how did I get on this soapbox...? ...no, I'm not going to ask you all to sing Kumbaya, I promise.)
That is the complete opposite of 5e's cleric, which rolls those "support" magics back into the cleric's main abilities. Curing disease, raising the dead, divining the future, You cannot replicate them non-magically
I think part of the problem is that you're focusing on how those things are done, rather than what they accomplish.
People recover from diseases and speculate about the future without the 'how' of 'using magic.' In a fantasy world, what might help a person recover from disease, or how long one might be *ahem* "only mostly dead," might be rather more open than it is in a scientifically-correct world. PCs in 5e D&D can be down and moments from death in a fight, and fine again an hour later, with no magical intervention having occurred, thanks to HD. Such extraordinary individuals might well be able to throw off a disease, or a disability like blindness, paralyzation or poisoning with some equally extraordinary motivation.
And, please, keep in mind, if any of that offends your vision of the world as a DM, you simply don't opt into the Warlord in the first place. It's a non-issue that some campaigns might not want to be that fantastic (while still having wizards casting Wish).
and other effects like this aren't separate (and available to any character with a feat or class ability); they're a main part of the cleric's identity.
Cure Disease -actually, in 5e, folded into Lesser Restoration - is on 5 different casters' lists. Raise Dead, on three. Divinations of various types are available to all casters. They can hardly constitute a main part of the Cleric's identity.
Which is why I suggested him more like a paladin; you MIGHT be able to get enough martial abilities to equal a half-caster with d10 HD and two attacks
What you're saying is that martial ability /must/ be strictly inferior casting. That's a legitimate thing to establish in a campaign, and D&D has certainly supported that kind of campaign in the past, and continues to do so in 5e. Introducing a Warlord who was the equal of a Cleric or Druid or the like would NOT make it impossible to continue doing so. It would hardly be a noticeable inconvenience in doing so, you'd just decline to opt into the class, using other, conveniently inferior, already extant options to cover such character concepts, which would have to be sidelines of a more traditional main contribution.
By the same token, though, adding the Warlord would re-enable styles of campaigns that were finally practical in 4e, for the first time (though they'd been attempted many times over the years).
but your not matching a dedicated "caster" is range or versatility without literally giving a warlord the ability to re-write reality nonmagically.
It's what heroes do - just not with a wave of their hands and a word of power - they change the world.
Warlords shouldn't be the /only/ non-supernatural heroes doing so, either. Hopefully the design space that might be opened up could lead to sub-classes less tied to the 5e vision of the Warlord, or new or other reprised martial classes, as well.