In the early jazz groups, the bass saxophone was the best bass instrument. There was no electronic amplification of the string bass, so it got buried by the ensemble. Some groups used a tuba, but the tuba has a somewhat nebulous attack on each note, and the bass line needs a definitive envelope of sound on the attack of each note to define the time. The bass sax did it without a microphone. As technology improved, things changed.
Also, most early jazz bass lines were 2 beat bass parts - The tonic and the fifth on beats one and three. Plenty of time to take a breath and even rest the chops for a fraction of a second. A good bass sax player could play forever.
When needed, the bass saxophone player could join the other horns to harmonize a line, or even provide a separate line. String bass was ineffective in this department.
Adrian Rollini, the virtuoso bass saxophone player, was also a concert pianist. He was completely familiar with the several roles his bass saxophone could play in any ensemble, and he demonstrated his skills on every recording he made. With the exception of Sydney Bechet, Adrian was the first jazz saxophone player. Any saxophone player who plays a bass line on his horn owes Rollini a debt of thanks.