High level fighters already regenerate like trolls and low level they can lay on hands themselves. So you already have limited cheerlord non-magical obviously magical healing. Battlemaster has a lot of the lazylord stuff already letting you give rogues another sneak attack every round. What exactly would you be getting by adding a warlord class? You can multiclass a paladin or bard with fighter and get healing and auras and superiority benefits and pretty much do whatever you want already.
Well, for one, you get the ability to play the character you want to play from first level, instead of having to wait several levels just to get the basic class.
Secondly, you get a class that actually functions as to what you want it to without a bunch of extraneous stuff (why is my warlord playing a lute? Why is my strategist a better swordsman than everyone else? Why is my coach lock tied to a diety and a code of ethics? Why am I casting spells?)
Everyone agrees you can get something that looks kinda like a warlord, eventually, in 5e. But, that's the problem. It's only about 70% of the way there and it comes with all sorts of baggage.
It's the same reason people wanted Psions to be their own class. And you can get a heck of a lot closer to a Psion with a Sorcerer than you can get to a Warlord with any number of multi-classes. But, people insisted they needed a Psion class, and they got what they wanted. So, yeah, I think Warlord fans absolutely should agitate for getting a warlord. Why not? Why should warlords get left out in the cold when psionic characters get support? it's not like psionics has ever been universally accepted. Lots of people dislike psionics strongly - for any number of reasons.
Actually, thinking about it, virtually all the arguments about Psions can equally be applied to warlords. Yet, Psions got a class.
Why not Warlords?