There are 5e martial characters that have resource management, so "if it's martial it has to be at-will" argument simply holds no water.
The best argument for a dedicated 5e Warlord class, at this point, are anti-magic fields.
Meh, it's just another example of how mechanics and fluff aren't fully separated in 5e. Meaning re-skinning isn't an adequate way of providing a significant past-edition option, like a PH1 Core class.
Which amounts to requesting that martial characters be even more versatile and powerful in a rare corner case of the game where every other type of character is effectively already neutered
That's a sad sort of spotlight balance, but it's what the very few non-magic-using sub-classes already shake down to, I'm afraid. It'd be cool if there were more to it than that.
(which is why anti-magic fields are an awful game mechanic 99% of the time and best used extremely sparingly for best effect and therefore are really not worth designing an entire class of character around).
The only edition that ever got away with that was 4e, and 'got away with' is maybe not the right phrase for it, since it was edition-warred against constantly for that unforgivable sin against then (super)natural order caster dominance.
I haven't read the whole thread, so maybe. Personally, I'm find with a warlord that has a lot of short rest and long rest abilities. Although I personally like them to have at least Commander's Strike style ability at-will, but I understand I'm in the minority on that one, even among warlord fans.
I don't know what would make you think that, the only appearance of the class so far was AEDU, it wasn't re-worked for Essentials or anything.
5e's class designs tend to work from concept to mechanic, so it's not a simple matter of plugging in appropriate/classic or flavorful abilities into a common structure. A Warlord could just have bland "you cannot use this again until you take a short rest" sorts of resources, but there might be more interesting/flavorful ways of handling it. A clever strategem, for instance, might not work twice against the same set of enemies - ever - but could be used against a new set of foes whether you'd taken a nap before encountering them or not.
There is still a small trend about: martial and non magical == at-will.
That started changing as far back as 3.0, but yes, it was part of the pattern of caster dominance (LFQW, 5MWD, etc) in the classic game and through 3.x - 5e has moved back in that direction, some, but no class depends entirely upon unlimited at-will spamming (even the rogue's SA is limited, just not in uses/time-period). Casters get at-will cantrips, non-casters generally some rest-recharge trick.
Another correlation driving caster dominance prior to 4e was flexibility. Martial classes tended to be locked in to their few choices at chargen (or, in 3.x level-up), while casters had significantly more choices, some available on a daily basis, or even round to round in the case of spontaneous casting or very flexible individual spells. That was why 3.5 prepped casters were so solidly in command of Tier 1.
The Fighter battlemaster is accepted with ressources managing, but there still hope for a pure at-will Warlord.
That would be pretty awful. The traditional D&D design paradigm places a very high value on always-available abilities, to the point that they end up being very limited in scope, power, and flexibility. And, the traditional D&D playstyles, which even 4e didn't deviate from in this particular sense, heavily emphasize resource management, particularly when it comes to support contributions, which directly bear on the pacing (encounters vs rests) that 5e depends upon to impose it's brand of class & encounter balance.
An all-at-will /anything/ is frozen out of the resource-management game, a second-class class, if you will. At-will support contributions, in particular, can't help but be either trivial, or at least theoretically 'overpowered.'
There are requests for martial manœuvres for everyone.
There are also complaints about at-will cantrip.
There's also past-edition mostly-at will classes that have yet to be reprised by 5e.
Indeed, but there was a lot of opposition to 4e regarding features whose presence are marginally unopposed in 5e. That's the issue. I don't want to delve into an exhaustive speculation as to why that may be the case. One simple reason, however, may stem from the difference in writing style and tone between the two editions.
You mean the more readable natural-language of 5e vs the more technical-manual quality of 4e?