Now if Biden had said said "I mean, you got the first mainstream African American running for president. He is articulate and bright and clean and a nice-looking guy", I can't imagine there'd be anything to object to.

Isn't it most likely that that actually would have been what Biden was meaning? After all, surely you couldn't get anyone who fitted that description better than Jesse Jackson did, when he was running for president? (As for "mainstream", Jesse wasn't that, because, in the circumstances of the time, there wasn't a chance he could win).