The thing about Jesse Jackson in the 1980s was, he wasn't just an African-American candidate; he was what passed for the standard bearer of seriously progressive politics in that decade--what passes for a Left in this country.
And when I saw the post title, with Jackson's name in quote marks, I wondered if that was what the OP meant--WI someone other than Jackson led a strong challenge to the Democratic mainstream from the left?
As I see it the whole problem of US politics in the '80s was that "vision," the mythic part of a political program that gives people a strong sense of what they are campaigning for, was largely monopolized by the Right. People had little positive sense of what they were voting for if they supported a Democrat, just what they were voting against. In opposition to the quasi-Randian ideology of the Republicans, wrapped up in red white and blue and talking tough, the Democrats could not articulate the different worldview in which the Republicans were not the only sane persons, despite what looked to me like overt and painful examples of Republican insanity that hurt the vast majority of citizens quite obviously and directly. (And fears that the same misplaced sort of cowboy rhetoric could easily lead straight to nuclear war and Armageddon).
Really this is a problem Democrats have suffered from at least since the aftermath of the 1972 election.
Jackson was a rare exception, someone who stood up and said what sort of alternative America he was for, one that addressed the bread-and-butter issues of the vast majority.
While we progressive types (I was one, being heavily involved in Jackson's '88 primary bid in Pasadena, California, as a volunteer canvasser) won few really impressive victories, we were a force to be reckoned with, enough so that I believe that Jackson got his traction as a sort of national figure as much from progressives finding an outlet and spokesman as from galvanizing African-American support.
Now, would just any African-American politician of any party have been able to do that? I suspect a moderate Democrat might have. Not a "moderate" Republican unless he so much repudiated the platform of Reagan-era Republicanism his status as any kind of Republican would seem absurd.
Even if we assume then that just any of these alternative African American politicians could have gotten exactly as many African-American votes as Jackson got OTL, it isn't clear he'd get as many non-African votes as Jackson did.