Two responses.
First, someone with an abrasive personality might still inspire by example. Or by force of will. Look at Nick Fury and his Howling Commandos.
Second, though, what makes you think we can hold personality constant and change class? Who thinks, for instance, that a classic paladin can have the same personality as a classic assassin? Or, in 4e, that an avenger and a barbarian would have the same personality? Or, in 5e, that a druid of the land and a Cthulhu-serving warlock would have the same personality?
Even if one treats class just as training, it's not in general true that every personality type is well-suited to every occupational type (eg some occupations require more patience than others).
Once one acknowledges that class extends beyond training to archetype, class and personality are clearly related in certain ways. Eg what how could it mean be true that a fighter of level X and 14 CHA is just as charming in personality as a rogue or bard of the same level with Persuasion expertise and 14 CHA?
This is where we have a collision of genre. (And maybe of politics, but that goes beyond the scope of the forum discussion.)
In Tolkien, for instance, a leader is not superior - and hence others are not inferior - simply in virtue of formal occupation of an office. A very important them in Tolkien is that of fitness for office (qv Feanor, Maedhros, Fingon, Thingol, the Numenorean kings, Sauron and the Ringwraiths, Theoden, Saruman, Eomer, Boromir, Faramir, Denethor et atl). A monarch or similar office-holder is called upon to prove him-/herself.
Hence capacity to inspire does not follow from office. Rather, genuine entitlement to office follows from capacity to lead and inspire. (This is all an elaboration of Bawylie's post contrasting Theoden and Denethor.)
That's a very pre-modern, Romantic European conception of leadership.
In what I would think of as a more American tradition, we have the notion of democratic leadership, grounded not in office (which may or may not be formally occupied) but from being the acknowledged first among equals. This is the model of inspirational leadership exhibited by Nick Fury, Captain America, and arguably even Conan among the Kozaks (though the latter is more complicated because it also draws in REH's views about racial and cultural superiority).
In the context of a RPG, the warlord doesn't need to enjoy formal authority to exemplify either of these tropes, or any sort of blending between them. The mechanics do make it true that the warlord is inspiring in personality (just as, eg, mechanics make it true that the bard is charming).
So you are saying that they are not inspiring because they are leaders, but they are leaders because they are inspiring. And since a Warlord is inspiring because the mechanics say he is, then he should be the leader of the group.
And this is where we really part ways.
The notion that inspiring someone is manipulating their emotions is anathema to me.
Yesterday, at work, one of my colleagues said something that made me smile and laugh. That wasn't a manipulation of my emotions. It was just making me amused for a moment. Another colleague complimented me for something that I said - that brought me pleasure, but it wasn't manipulation.
The inspirational warlord rouses his/her companions spirits. Restores hope, when it is flagging. Reminds them of what is at stake, and why it is worth struggling for. Etc. None of that need be manipulation (and, in the context of the genre tropes, is not - the cynical, manipulative Nick Fury of, say, the Avengers movie is not a warlord in the spirit of these tropes).
If one can't think of the emotional dimensions of human relationships and interactions except via the notion of manipulation - which is the atomistic conception that I have referred to upthread and in other threads - then of course there is no room for the warlord. And presumably, on this model, the cleric and paladin are just variant wizards or fighter-mages.
But there is nothing in 5e that confines it to this conception, and leaves no room for those other, staple, genre tropes and archetypes.
And I don't see why this ability to make people smile, or to inspire people, or to encourage, should be limited to one class. My Wizard can make people smile. My Fighter can inspire people and my Rogue can encourage people. I fail to see the need to give a mechanical benefit to things like this. It is already possible in the game for any character.
And I don't see the need to require other characters to be inspired or encouraged by one particular character because it says so on his character sheet.