I'm not talking about personality, although I did use that word in my query
Oookay....
I'm talking about inter-party relations. I keep hearing that mundane inspiration can work no matter how the rest of the party relates to the Warlord.
Sure, there are different ways someone might be inspired. Their relationship or lack of one to the person trying to do so would have a bearing. You might be inspired by a speech given by a politician because you're both patriots, even though you don't agree with many of his policies. You needn't see him as a larger-than-life hero, either, he can have feat of clay and still get something across to those listening. You might be inspired to greater efforts by a rival in an attempt to out-do him, or prove you're more capable than he's snidely suggested. Superficially similar, but with a different underlying relationship, what look like insults could actually be encouragement ('good natured ribbing'). You might be inspired by a 'lead from the front' hero doing great deeds that you can barely hope to match - or by a much less capable ally trying his best to keep up with you. You could be inspired to rush to the aid of virtual non-combatant in danger. You could be inspired by the honor of serving a social better, or the duty of aiding a social inferior.
That said, and for what it's worth, I think classes come with certain baseline implications about personality, but there's still a spectrum of possibilities for each one.
Some classes more than others, but even then only about some specific traits, not the whole personality. A Cleric can be expected to be pious, a bard outgoing, a wizard studious, sure. But a wizard could be introverted or arrogant, obsessive or circumspect, seeking knowledge for the sake of power or power for the sake of knowledge. Heck, that most classes are open to any alignment illustrates that they don't much constrain personality or RP choices. The Warlord didn't have any alignment constraints, and we've discussed wide variety of possible ways that just an Inspiring Warlord might've done his thing - and there were a number of other builds, as well.
If you think there's no way Boromir could be a Warlord, why not?
Classes are mostly narrow enough in competencies that it'd be hard to fit most fictional characters into only one of them.
And do we agree that at most tables, the rogue or bard would be discouraged from using the Persuasion expertise on another party member?
I think this is where the line between player and character becomes an issue. Players try to persuade eachother, as a matter of course, with all that entails, and players do, ultimately, make decisions for their characters. So a player who is on board with an idea, but feels his character wouldn't be, could use a persuasive ally as a rationalization, for instance.
This is where I see the bigger problem, actually. I would venture to say that most players of D&D come from cultures that like to think of themselves as meritocracies. We like to think that positions of authority are granted to those who display the greatest capability to lead--and if they are not, then an injustice is being done.
We also eventually come to know better.
And, most D&D characters are from a medieval society, so, oh well.
So when a player shows up with a character concept whose entire heart is "I have a marvelous capacity to lead, better than anyone else in the party, and it doesn't even come from magic but just from my own natural qualities," the obvious corollary is "Therefore, I deserve to be placed in a position of leadership, and if I'm not, then an injustice is being committed."
Most PC classes wield remarkable magical powers, any of them could claim that as marking them for greatness. "Check me out as a Channel Divinity and even Raise the Dead, clearly the Gods have chosen me to be your leader!" And, of course, simply picking a Noble background could give you legitimate - within that culture - authority.
So, no, even if that extreme hypothetical, which is at odds with the explanation of both the class and the 'leader' role from the Warlord's PH1 appearance, you're not pointing out a problem unique to the class, but a problem you have with a sort of character concept that can already be pulled in a variety of ways in 5e, if one were so inclined.
But what do you say if I want to be able to decide for myself whether what another character says is something my PC would find inspiring?
You can't hear what that other /character/ is saying or how he's saying it, because the interaction isn't really happening. You have some game mechanics, and a player and/or DM description to go off.
I hope it's more than "Just decline the buff," because while that's a possible solution, it doesn't strike me as an ideal one.
It's a complete solution, and arguably more flexible/accommodating than analogous mechanics for a variety of existing abilities in that regard.
What would make it 'ideal?' Someone tries to help you, you can accept that help or decline it. What's missing?
would it be somehow against the concept to play a Warlord like Nick Fury
The mechanics could cover quite a range of concepts, so it might be contrary to one concept, but well within another. Maybe he just didn't like the 'manipulative' version of Nick Fury, for that matter.
Which means that I struggle to see how the opposition to the class is one to the pieces that make it up rather than being a direct opposition to the class on grounds I can only see as incoherent.
Sometime the point of opposition, is opposition, itself.
Can you go into more detail about the "comic relief inspiring warlord"? This is the first I've heard of that concept. Would this be like, say, Jar Jar Binks accidentally killing battle droids in The Phantom Menace? (Note, I say this as someone who actually likes the Star Wars prequels, so that comparison is not intended as a slur.)
Another thing we can agree to disagree about.

But, given that PoV, sure, that could be a way you could imagine, say 'lead by example' sorts of mechanics. Or, for that matter tactical ones, especially the reactive sorts, your bumbling PC inadvertently distracts enemies, leads them out of position by being a seemingly easy, low-risk target, and thus sets them up for his allies. Oddball, but too much of a stretch, nor entirely contrary to genre.