Do you also think that Westergaardian theory is superior for understanding jazz?

Yes. My claim is not repertory-specific. (Note that this is my claim I'm talking about, not Westergaard's.)

More generally, I claim that the Westergaardian framework (or some future theory descended from it) is the appropriate one for understanding any music that is to be understood in terms of the traditional Western pitch space (i.e. the one represented by a standardly-tuned piano keyboard), as well as any music whose pitch space can be regarded as an extension, restriction, or modification of the latter.

I've encountered jazz pianists on the internet who insist that harmony and voice leading are ABSOLUTELY ESSENTIAL for doing jazz improvisation and anyone suggests otherwise is a heretic who deserves to be burnt at the stake.

How many of them are familiar with Westergaardian (or even Schenkerian) theory?

I've encountered this attitude among art-music performers as well. My sense is that such people are usually confusing the map and the territory (i.e. confusing music theory and music), à la Phil Goetz above. They fail to understand that the concepts of harmonic theory are not identical to the musical phenomena they purport to describe, but instead are merely one candidate theory of those phenomena.

jazz classes do seem to incorporate a lot of harmony and voice leading into their material and their students do seem to make fine improvisers and composers

Some of them do -- probably more or less exactly the subset who have enough tacit knowledge not to need to take their theoretical instruction seriously, and the temperament not to want to.

Oh, and for what its worth, you've convinced me to give Westergaard another shot.

I'm delighted to hear that, of course, although I should reiterate that I don't expect ITT to be the final word on Westergaardian theory.