Why shouldn't the Warlord be able to fight well?
Not an issue, really. In 5e, everyone can fight pretty well. You get HD to weather fights and weapons or cantrips (or both) for offense, and
everyone gets the same proficiency bonus. In a basic competence sense, wizards fight as well as fighters. An S&B fighter and an abjurer wizard with the same DEX, playing darts at the pub, will be about evenly matched in terms of accuracy. In the practical sense of who wins duels when magic's not involved, the fighter, however, is "Best at Fighting."
There are only a couple of D&D concepts that really call for outright non-combatant status: the Pacifist Cleric and the Lazy Warlord, for instance.
The other warlord concepts don't call for it to be the best at fighting, either, nor even second-best like a Barbarian or tied-for-best-if-I-can-use-magic like a Paladin, nor placing like the Warlock or Rogue by virtue of DPR, nor showing like the Ranger. Well, the Bravura maybe should 'show' in personal combat - maybe get a choice of two or three melee combat styles and an Extra Attack at 5th (really, it's the only Warlord flavor that would be plausible as a Fighter sub-class). And, the Defender/Protector/(lifeguard ;P )/etc that popped up in this thread maybe should have, well, Protection or Defensive styles.
I think there is a difference between what a warlord does and simply having an entourage of followers.
The difference is features that support the concept. The old-school fighter as 'Lord' is like a magician class that has all sorts of mystical trappings and flavor and automatically gains a respected 'court magician' position at 9th level - but doesn't actually know how to cast spells.
Caesar, for example, strikes me less as a "figther" and more as a "warlord." I don't think he necessarily would have been a "high level fighter," but, rather, a moderately levelled warlord. But I also think that a lot of mytho-historical figures that I think of as warlord also come out of literature such as Romance of the Three Kingdoms.
One of the problems with modeling characters from myth/legend, literature, and especially history, in D&D, is that D&D so tightly links competence to level, so if you were really world-class at something, you had to be high level, and thus had to be a beast in combat. If you were world-class at anything remotely martial & didn't use magic, you fell into fighter, for lack of anything else, and had no choice but to be a tank. Particularly good at strategy & tactics, or a natural leader & great public speaker, but without magic, shouldn't map to "Best in personal combat, before magic - or strategy or tactics - come into it..."
The 3e Sorcerer says hello. The 3e Warlock also says hello.
Yes, there have been a lot of classes born /just/ of mechanical necessity/experimentation, like the 3.0 Sorcerer, 3.5 Warlock, Warblade, War Mage, (really anything starting with War it seems like), Ardent and PrCs like the Mystic Theurge among others, and like the 4e Avenger, Invoker, and Warden.
There have also been those born of just a concept that, while do-able mechanically, were excluded from the best ways to do it by over-narrow fluff. The Witch might be an example, a wizard could be a witch, but all the Vancian stuff didn't fit. Or a non-LG holy warrior.
Then there are those at the happy intersection of mechanical & conceptual 'need' (more like 'nice to have' it's a game, it's not like we /need/ clerics or wizards, either), like the 4e Warlord & Shaman, the post-Essentials Skald and Elemental Sorcerer, the 2e CHP priest, the 3e Artificer, etc...
But, regardless of the genesis of a concept, it becomes a valid concept, for D&D, just by having been there. Especially in the context of 5e.