The word 'jazz' is an umbrella over so many disparate styles that it has lost meaning in the same sense that 'popular' music could mean anything or nothing.
Jazz embodies a tradition of improvisation, although most improvisations have harmonic formulae and configurations that have been fleshed out many times. If improvisation is the issue, then among the greatest public improvisors in history were Mozart, Beethoven, Chopin, Liszt, etc. It was a foregone conclusion that great pianists of the 18th and 19th century would improvise for their audiences.
The tradition of not improvising is comparatively recent; midway through the 20th century. Through the years I have run across various written accounts of Rachmaninoff, Hoffman, Paderewski, and others improvising as a prelude to the printed program. The irony is that improvisation is actually more in the classical tradition than so-called 'classical' piano recitals of today.
Jazz piano teachers I know are frequently more conscientious in teaching harmony and theory to their students than many classical-only teachers I have encountered. Yesterday I took on another new student who had been studying with a 'distinguished' teacher at the Mannes music conservatory here in NYC - according to this new student, her teacher refused to teach her scales because "the scales are already in the music".