There's no shortage of character archetypes and roles that could be denoted as classes. Some broad and some small. Anything could be a class if you wanted.
Nod. But there's only so many that have made an appearance in a D&D PH1. The Warlord is one of those.
But just because something is a trope, an archetypal character, doesn't mean it needs to be a class.
Classes are how 5e delivers most of a character's capabilities, so the 'need' is really only there when mechanical support for the concept isn't already available from other classes. Since 5e doesn't assume MCing or Feat, BTW, so 'other classes in combination' doesn't cut it, either - thus, for instance, we have Paladins even though a Fighter/Cleric could fit the bill, and an EK even though a Fighter with Magic Initiate might squeak by.
The Warlord concept requires mechanics that don't exist in sufficient number/power in one class. MCing+ Feats + Backgrounds could get pretty close to suggesting the concept, though not as close as a Fighter/Cleric is to a Paladin, and such a build wouldn't be terribly viable if it stuck to doing concept-appropriate things.
The barbarian, paladin, and ranger totally do overlap with the fighter. As does the monk really. Ditto the sorcerer and warlock with the wizard. Really, you only need three classes: the warrior, the expert, and the spellcaster. Everything else is flavour.
Even splitting out warrior and expert is just a matter of cleaving to tradition. Warrior is all combat, expert almost all exploration/interaction - you could combine them without making the result in the least OP relative to the caster. So, really, Hero and Caster could cover things. For that matter, classless systems do a fine job of covering every conceivable concept, anyway.
But stuff like the cleric and ranger get included because tradition. They get grandfathered in because they've been a part of the game since 1st Edition. They've been included for two or three editions and a couple in a row.
And the Sorcerer, Warlock, and Warlord have been in only 1 PH1, yes. 5e was meant to draw from all editions, not just the older ones, and to be for fans of all editions, not just for h4ters.
They're not just fantasy tropes, but D&D tropes.
The cleric is arguably not a fantasy trope, at all, the Vancian caster certainly isn't, FWIW (nothing, in D&D). But, yes, 5e tried very hard to feel like D&D, and that necessarily meant including extraneous and redundant material from the early days of the game.
The criterion Mearls asserted was 'in a prior edition PH1' - all the '1' did was exclude a lot of 3.x and 4e classes, since the few AD&D classes not introduced in the PH (like the Barbarian) were helpfully in later PH1s. The Sorcerer, Warlock, and Warlord were the only classes to appear in only one PH1, the Warlord and Warlock both only in the 4e PH1.
Even then "person who leads people but isn't overtly magical" is a pretty weak archetype. Because it's so very, very easy to have a leader that's also a wizard or paladin.
Actually it's a very powerful (in a storytelling sense) archetype. And one much more prevalent in genre than any sort of magic-wielding protagonist.