Players in jazz swing and big bands often use sheet music as well. The more complex the music and the more people playing simultaneously tend to make having sheet music more reliable than memory. Most studio work is off something in writing, whether its lead sheets or full transcriptions. Saves rehearsal time and wasted energy to have written music.

All of your points show an absolute ignorance of jazz. Jazz has been around for over 100 years and I can find artists in every decade that would disprove all of your points. From Louis Armstrong and Duke Ellington to contemporary artists like Ezra Collective and Moses Boyd there are plenty of examples, of course there are also extreme examples that do meet your criteria but that is not all there is to jazz just liked Norwegian Black Metal isn't all there is to rock.