Really interesting to see Paladin and Bard as the leaders currently. I might be misunderstanding the question, so I'll start with how I think they'd work as subclasses and then my opinion on if its a class of its own.
Seems really possible to do a Cleric domain that has a
little warlord in him (as the War Domain does) but I'd want more. Paladin could work similarly.
Fighter and Bard seem like they give the most blank slate to allow stuff to make it a Warlord subclass.
If we're talking about a full class, I actually think starting the subclass at level 1 in a way similar to the Cleric could be quite good. Use the same types of Warlords we had in 4E as the "domains," maybe add something new as well. Channel Divinity could be used basically as it is, with a new name. In place of spells, you add something like battle master maneuvers that are focused on buffing allies and granting attacks in certain scenarios.
Reading about others talking about the Monk as the chassis, I agree that this could be a good chassis. Ki-driven (with a new name for ki) but focused on buffing and granting attacks, with some sporadic bonuses for movement and maybe one ability to help initiative for an ally (but maybe not to the point of initiative tricks we had in 4E).
I think people may vote for paladin for their Aura and the bard for the inspiration die, which are indeed interesting mechanic that could be modified for a martial support.
I figured bardic inspiration but I hadn't considered the Paladin's auras. Good point. This also reminded me of the Sentinel Druid and the Fey Beast Tamer theme from 4E which had auras that gave a small benefit while allies were in it, or a buff to attacking enemies that were in it. I think exploring the aura concept further would be an interesting design space for a warlord rework.