So, assuming that you have 15 levels worth of other classes already existing to fill in that overlap, why wouldn't warlord fit into an prestige class?
That's quite an assumption. I'd be stretching to credit that there are even 5 such levels.
And, Prestige classes are just very narrow, conceptually. Borderlands Marshal might be a PrC, for instance, with a requirement that you hold a keep on er, the borderlands... Just an example.
So, could you do the Warlord with a 'Warlord' PrC? No, not even close.
With a dozen or so 5- or 10- or 15- level PrC's? Maybe, but most of what you could cover that way would be covered as well, and more efficiently, with a full-class Warlord, the odd new Background, & MCing. And the full-class Warlord could cover a lot more, besides, including being able to play the character you want from 1st level, and enabling styles of play that require a non-magical support class, again, from level 1 (and, really, 1st level is probably the time you need a support character the most critically).
I'm not saying it couldn't make a good 20th level class, this isn't an either-or.
I think that there are Warlord concepts that could be done with the full class, but work even better if there were specific PrCs for them in addition to it. As long as it's in the context of a full class, warlord-focused PrCs are a great idea. Ideally suited for concepts that eventually demand high rank (like General or Marshal) in a military hierarchy, or political authority in a region, or the like.
What are the salient points of a warlord in 5e - I think having some sort of idea what we need to fit in would be needed.
I believe there are a couple of threads already going into that. There's past concepts and builds it needs to cover, there's being adequate as the party's primary or sole source of support contributions (thus enabling low-/no- magic campaigns & all-martial parties and generally adding a unique support choice to broaden the potential appeal of that critical function), there's being a worthy successor to the original (to avoid the appearance of excluding it's fans, if nothing else), there's aspects of the concept compromised in the previous version for the sake of design-philosophy boundaries that can now be crossed in 5e...