Exactly like concerns over the name of the Martial (source label) Leader (role label) Warlord (class label). As has been pointed out pretty nearly the whole run of D&D, characters don't necessarily take their class name as a job title or descriptor in-character (I suspect level-/Titles/ may have gotten used in 1e, at times, I know I went with 'em a few times, though it was funny the way your magic-user changed specialties - Theurgy & Thaumaturgy, FREX, very different outlooks - or the Cleric changed
religions if you took those literally, too).
The stronger point is that a position of legitimate or granted authority, like a Captain, Commander, Marshal, or the like, and the quality of Leadership that helps others work together and put in their best performances are quite separate things. When we say "but any class could be The Leader," we mean any class (or level) could be put in charge arbitrarily, or be making decisions and giving orders. But the skills & talents of leadership needn't come with that, and needn't be exercised from such a position, a second-in-command or trusted follower could be the one exercising such skills, an Aaron to the chosen leader's Moses, as it were, heck sergeant who bucks up his lieutenant in a tough spot could be exercising leadership not just on behalf of his leader, but /on/ his leader. The actual leader still needs to come up with good decisions, goals & objectives, but the leadership can come from elsewhere.
And, for better or worse, focused, exceptional, high-impact, but non-supernatural skills & talents like that, in D&D are modeled by a whole 'nuther class separate from the Fighter. From the Thief and Ranger, back in the day, through Cavaliers, Bandits, Experts, Knights, Duelists, Scouts, &c, on to the Warlord in 4e.
As inclined as I'd be to give a lot of Tolkien characters some levels in Warlord, the Fellowship was openly led by Gandalf until the bridge, and, even then, he gave them a last command before falling into the abyss.