If you'd been born with the skin colour referred to as "black", John Hardly, that wouldn't necessarily mean you were African-American. After all most people with the range of colours referred to as "black" aren't. They (or their ancestors) might come from Africa, India, Australia, and quite a few other parts of the world. And of course there are many Americans with dark skin who don't have African ancestry.

If you wanted to indicate that your family of origin came - at least in part - from Africa, and not from those other parts of the world, and ended up in America, the obvious term is African-American, set alongside Irish-American and Italian-American. Nothing "asinine" about any of those terms so far as I can see.