It surprised me that no one took the position that it is not necessary to read to play jazz. I play with an awesome piano player every weekend. He sounds like a recording artist but can't read his way out of a paper bag. The trick is, he knows several hundred tunes that he learned from records. Remember, jazz was a folk music before it was accepted in academia.
No. Jazz is an opportunistic merging of folk and "legit" music. Many of the founding figures in jazz could read. Many of them eagerly studied classical music when they got a chance. Some of the styles that jazz emerged from, such as ragtime, had widespread published literature. Fats Waller published jazz tunes in the 1920's. Jazz had already evolved to the point where most jazz players could read, by the time that jazz gained acceptance in academia.
The OP didn't specify what kind of jazz band. The first thing that comes to mind when someone says "jazz band" is a big-band. I have played with roughly a dozen different big-bands over the years. Every single one of those bands had written charts. I could not have functioned in any of those bands without being able to sight-read. Big-band charts cover the gamut from chord charts to written-out bass parts with no chords, and mixtures of both formats within any given tune.
Most of the players in a big-band come from a school band training, where everybody learns to read as a point of departure. Many of them aren't even aware of the guitar-oriented world where people learn to play without reading.
Combo jazz is of course a different story -- players have varying levels of reading ability. You can get away without reading if you know tunes, have a killer ear, or can call which tunes are played. Players who are strong readers get a reputation for it, and get the calls where reading is required.
So I will agree that it's not necessary to read in order to play jazz, but at least in my locale, there are enough bass players who can read, to cover all of the gigs.