I’m not American; never been to America. So I grew up with different culture. The dark skinned ethnicities near me were mainly Pakistani, and I don’t remember if they were happy to be called black or not. I think we basically grew up feeling like you have to ignore skin colour, the same way you ignore the size of someone’s nose. We weren’t supposed to see it as any more different than someone else is from Wales, and someone else is very tall, and someone else lives in this or that neighborhood - but to comment on ‘black’ skin or big nose might give offence.
I agree ‘African-American’ is an awkward term also, as you say.
I suppose part of the difference is the black community in America, as I understand it, has a very strong cultural identity, whereas when I grew up the idea was basically that your ethnicity was another part of your background, but not your community identity. A British Indian is a Brit who happens to have Indian heritage, that they may like to hold close or may like to distance from: but we’re all British. And someone from South Kensington might talk all posh an’ all; and a Scouser’s gonna Scouse: but we’re all British. That sort of thing. (And if you’re not British we still welcome you just as fondly; and to do otherwise would also be racist.)