The character works fine with others, except the Bard and the Warlord.
That's just begging the question. You hypothetically decided that your character would be uninspireable for whatever reason, refusing to be inspired and foregoing bonuses from Bard and Warlords and Clerics casting Aid and maybe even refusing the occasional Help action depending on how it's described, is all just supporting that character concept, mechanically, not sabotaging it.
If you don't like those types of characters, fine, don't play them. I don't like Bards, so I don't play them.
I can't pretend to disagree with you on that point. ;P
Inspirational classes require all other characters to be inspired by them. One based on team tactics would require that the other characters accept their tactics, and, yes, magical characters require that the other characters accept their magic.
All characters 'require' their concept to be acknowledged by the other players around the table, and chargen mechanics are a way of resolving those concepts so everyone can have a remotely compatible idea of what everyone else is playing. Bards and Warlords are in no way unique in that sense. Any character concept become an issue of someone at the table feels a need to deny it.
That potential, even if you think it is greater with some classes than others, is no reason to deny a class or concept to /everyone/.
*So if I don't try to stop people from playing Bards, why am I trying to prevent Warlords? Well, I don't want Bards either, but they are already in the game, and have been for some time.
That's no different than saying that the Warlord and it's fans must be excluded because it was from 4e, and that's very much at odds with the spirit in which 5e was conceived.
If you want to dictate to people what they can and can't play, do it to their faces, at the table - don't try to get WotC to do your dirty work for you.
Sure, you can just decide the character is inspired and everything works fine, but that's the thing. You have to decide that your character is inspired for everything to work. Having your character not be inspired if a false option. It's only an option if you are willing to refuse a buff, reduce the fun of the other player and be accused of lacking teamwork (which I have been accused of in this very thread).
It's a very real option, it's just an option that has mechanical support, and may open up RP opportunities, depending on what it is about your character (assuming it is about your character) that's causing the issue, and how it might be resolved.
And it is understandable that people who play Bards and Warlords would never notice any problem. That's because people like myself don't say anything during the game. We don't want to disrupt the game or bring anyone down. So we bury the annoyance and move on. Then, in my case at least, it picks at me more and more until I can't take it anymore and I have to express my annoyance to a bunch of strangers on a web forum.
Don't kid yourself: everyone goes through that about some pet peeve or other. Awful-Good Paladins. Psionics. Kender. Meta-gaming. Threats to immersion. Class imbalances. Hit points. Vancian casting. Gnomes. Whatever. Burying annoyance isn't as good (for you) as letting it go, or even opening up to the possibility that there's some positives to the thing that annoys you, or, at least, being honest with the folks you're gaming with about what you want from the game.
But, sure, venting about it on-line, no problem. Demanding WotC take your venting as reason to exclude options from the game, OTOH, not cool.
As I've said, I've even played in a 4E game alongside a warlord and not given it a second thought. But in retrospect, I do dislike having to picture my character as worshiping the warlord's wonderfulness every time she handed out bonuses.
There's never been any need to do so. That's only one possible way of imagining inspiration, and a pretty hackneyed one, at that. And, you didn't even do it at the time, it's only in retrospect, when you go looking for a problem, that you've fabricated one.
