Generally, you don't play the bass to become a star.
While taking bass lessons in high school, I was pleased to see Chicago's "Colour My World" in one of my music books. My bass teacher gave me clearance to learn it, but practicing it brought home the dissatisfaction a bass player sometimes has. Yes, I was playing "Colour My World," but it was the rhythm part, with no hint of the melody. No one hearing me play it would have thought, "Hey, that's 'Colour My World.'"
On the flip side, Henry Mancini's "Peter Gunn" lets the bass player in on the main theme, and it's a catchy one. A saxophonist can jump off the stage, walk out into the audience, and wail away on that theme. The bassist, however, has to stay on stage, playing the same riff the entire time. While it's the riff that propels the tune, it feels odd and repetitive to play just one lick over and over.
That's the nature of the bass, though; except when playing a bass concerto or recording your own album in the style of Ron Carter, Stanley Clarke, or Victor Wooten, a bassist's role is a supporting one.
