If bardic Inspiration doesn't cause that disconnect you speak of then neither would a warlords inspiration based abilities.
The point you're missing is that bardic Inspiration is
maaaagic and therefore we can completely ignore any disconnect.
I mean, if you can ignore the idea that the follower of two opposed dieties can affect each other with beneficial spells with zero consequence, then, ignoring magical singing is pretty easy.
Of course, it all falls apart if you drill down even a little because it makes zero sense for a Paladin of Bahumat to accept the blessing of a priest of Wee Jas. But, then, treating clerics as just like wizards is pretty much par for the course - the paladin isn't accepting anything, he's simply the target of a magical effect, no different than a wizard casting Haste. The fact that it's divine magic, comes from a divine source and that divine source is intelligent, aligned, and comes complete with a belief system, doesn't matter. A spell is a spell is a spell. :/
Me, I find it utterly baffling that we can accept flying barbarians, teleporting paladins, and fighters that can grant you temporary HP after hitting an enemy, but, apparently, a warlord is a bridge too far.
If you have an issue with warlords granting buffs or additional actions, how do you justify Battlemasters in your game?
See, this is the whole problem in a nutshell. EVERYTHING that a warlord could do in 4e can be done in 5e. It just can't be done by one class, nor can it be done very often. But, any criticism you can make about 4e warlords equally applies to 5e. It's so bizarre. "Warlords create this disconnect ... but .. I have no problem with a stranger smashing an orc and granting me a d6 bonus temporary HP."
I just wish those that have an issue with warlords were actually consistent in their arguments. How can you possibly bitch about a warlord but accept a Battlemaster?
How is my Mastermind Thief granting you advantage on your attacks every round? Why don't people have an issue with this? I can do it at range - so, for some reason, my complete stranger, standing beside you archer, makes you a considerably better archer, every round, all without magic. And this is okay, but a battle master is bad? WTF?
The only thing I can think of is this is just edition warring with a funny set of glasses and a fake moustache.