Yeah, one can't ignore the differences in public opinion on civil rights between '52 and '60 that easily... Sparkman may have been tolerated in '52, but those eight years made a difference outside of the Deep South - definitely unacceptable by '60 standards. The fifties began the age of TV, and that played a big role in the early civil rights movement. Specifically the Till murder, which was a difficult thing for anybody to stomach, racial preconceptions notwithstanding. People could see when something isn't right, and despite political tensions in South surrounding desegregation, Southern Democrats seeking to rise in the ranks as a national figure needed to play a more rational line.
As bguy stated, even the moderate by southern standards Kefauver didn't turn off most of the Solid South. A '60 version of Sparkman might excite the Southern Manifesto crowd, but would be less warmly received among the middle class in Illinois, New Jersey, Michigan, Pennsylvania...
Agreed that Gore Sr. could be a possibility.