The reason I ducked is that this is a variation of the 'does the lacquer affect the sound' sorts of threads, which are quite controversial.
Yes, different material sounds slightly different. Tim Price, and a few others play on specially crafted wooden necks.
That being said--your sax playing friend is right. At a certain level it's the archer not the bow.
If you want to modify your lady, by all means go ahead--she's your lady. But , IMHO there's not going to be a significant difference in the sound.
I'm in the camp that believes that it's the geometry of the bore of the horn that most affects the sound, not material or thickness.