Yes, pressure changes will produce significant changes in pitch, added to the variation in volume, and enough to produce the unpleasant vibrato (or tremolo) characteristic of some old singers. One may try to sing a note while pressing rapidly one's stomach, and will see that it is very difficult to keep a constant pitch. My teachers agree that a good technique to reduce that kind of "voice trembling" is to alternate "crescendo" and "diminuendo" (growing and diminishing) in each note or phrase, thus avoiding a constant flow of air, which is so difficult for some people. (I am a trained amateur singer but not an expert)

Very interesting what you explain about instruments, I'm sure it takes a lot of study to obtain that sound we admire.

Un abrazo,
Andrés