Mike, most of the models about woodwind sound assume that the walls/body are sufficiently thick/rigid not to be deformed by vibration of the air column.
I recently read an article by Ralph Morgan about how mouthpiece construction affects it's vibration and the sax sound. Ralph Morgan was a highly regarded mouthpiece maker after he retired from Selmer USA. What he says is that making the mouthpiece thin/light enough to vibrate will change the sound. Afik this is empirical, not measured.
https://www.dornpub.com/SaxjPDF/Material.pdf
I've read in many places that the neck has a big effect on sound, due to it being narrow and early in the sound making. Does its vibration affect things? No idea. but it gives some substance to claims about nodal weights that guys like me dismiss as snake oil.
But remember, the sound vibrations are longitudinal, not lateral. So tube/wall vibration is not in the plane of the sound generating vibrations.
I read things this way, but may well be wrong - it's my understanding gleaned on a lot of reading:
Saxes are solidly constructed and the wall vibration makes a marginal, inaudible contribution to sound.
Sound generation is mostly down to player, followed by mouthpiece/reed/lig
Airflow through the sax body is negligible and has no little/no effect on sound.
The mouthpiece is a complex interaction of embouchure (including complete vocal tract/lungs), reed, chamber shape, material which converts pulses from laterally vibrating reed into pressure cycles along the length of the sax body. Thus mouthpiece vibration may well affect sound by acting on the air in the mouthpiece as it's movements transform into the longitudinal vibrations further down the tube.
The neck is rigid, doesn't change shape from the vibrating air column inside it so movements do not affect sound.
These interactions affect the mix of the different partials which make up the sax sound.