Yes, but nobody in this thread stated otherwise.
I've seen a lot of Americans refer to something as being "British" only for Brits to leap on their backs and say "Aha! There is no "British" accent!"
But the American as often as not was simply referring to an accent that was British, they were not saying all British people have that accent.
It depends. Some are aware there are other accents, others when learning I'm English launch into some weird lah-di-dah piss-taking accent (or its typed equivalent) that's utterly unrelated to the way I speak. But I get what you're saying, in much the same way I'll describe someone with an American accent as having an American accent, even though I'm aware that there are lots of American accents. I think it comes down to context, in this case "[I dislike] children with British accents" which seems oddly all-encompassing to me; I mean, I dislike the whiny brats in Skyrim but it would make little sense to say "I dislike children with American accents".