I suppose there "might" be a way to do that and not have it appear to be racist, but I have a feeling your assessment is correct, that it did not come off that way.
Jolson based his act on the Minstrel show style, which was a truly racist form of theater that used white actors in blackface to perform as stereotypical black characters telling jokes & stories. There was always a white-face "interlocuter" between them to feed them the lines. Many times, one of the blackfaces was a country bumpkin and the other one was a cityslicker. At times, it could be a way for the black actors to poke fun back at the white guy, the comedy went both ways, but obviously, the whole concept was born out of stereotypes which are no longer acceptable in our society.
Al Jolson was originally a minstrel show performer in Vaudeville who later took his act to the first talking movies. "Mammy" was one of the first film recorded musical performances. For that, it has historic value, but not much for any other reason.
I am surprised Disney would do this.
History on Minstrel shows here: http://www.musicals101.com/minstrelb.htm