Well, I actually don't think that's a great reason. We still say someone is Irish, even if their family has been in America for a long time(note, I agree with you that we shouldn't, I'm just saying we do).

However, there's a better reason not to call a black person "African-American". That is that many black people actually came from places other than Africa, mainly from the Caribbean islands. Someone from that area isn't African-American at all. It'd be like calling all white people European-American, when some of them might be from places other than Europe.