The interplay between persona/projection/charisma and the music itself is always complicated. In the case of Prince, the music is both collaborator and counter-foil to the gender ambiguity of his look and style, the contrast between his stage presence and his reclusiveness and the tension between his Jehovah's Witness-straightness and his sexual explicitness.
These kinds of tensions were present in the work and very public lives of Ray Charles, James Brown and Michael Jackson. However, the cultural impact of these three resides more completely on the bedrock of their music. Prince is reckoned to have done everything supremely well; everything being the key word. Time will tell us if his eclecticism begat something musically new and reproduce-able, or if his influence will ultimately derive from his persona.
In the case of jazz, media saturation has always been quantum levels lower, especially for black musicians, and the paradigms I describe above were unlikely to play out as publicly. Still, there are parallels to be seen in jazz careers. Below are five important figures in jazz and brief descriptions of how I think the personal and the musical interacted to determine the scope and area of their influence.
W.C. Handy: His compositions, chiefly St. Louis Blues and Memphis Blues, were widely performed; he organized an orchestra that hovered between ragtime and jazz and he did have some influence within the world of popular music. However, his organizing and entrepreneurial skills brought him much wider cultural renown, to the point where he is widely known as "Father of the blues;" a phrase that both overstates and misplaces his musical importance.
King Oliver: A trumpet player who was influential musically in the late 19-teens to mid 1920's. You might liken him to Sidney Bechet in that respect, but unlike Bechet-a strong, sometimes volatile character who carried on for many years-Oliver's health issues, a lack of personal charisma and business naivete greatly shortened his career. Oliver's wider cultural impact has been largely relegated to "the man who brought Louis Armstrong to Chicago."
Duke Ellington: His work remains a perennial influence in jazz (not a word he cared for), but he has achieved wider cultural renown. Aside from songs and jazz compositions for his orchestra, he wrote film, television and sacred music and was compared with America's best "classical" composers. His persona is relevant. Ellington seemed perfectly comfortable performing for the rabble and for royalty and his elegant and somewhat enigmatic personal style had a lot to do with bringing him wider cultural acclaim.
Charlie Parker: The co-creator of Bop presents an interesting case. The jazz community acknowledges him as arguably its most influential musician. During his life, he was acknowledged by members of the wider cultural, non-jazz elite as an artist of the highest calibre. Yet, while his name took on a meme-like character ("Bird lives" graffiti) and many in the non-jazz community may say they have heard his name, the trappings of wide cultural renown aren't there. What do I mean? Streets, schools and scholarships very rarely if ever, carry his name. Chic chefs, fashion trend setters, politicians, advertisers and mainstream media seldom, if ever, refer to him as a cultural touchstone. Had his drug use not been so widely known, his place in the wider culture would probably be very different.