I think there is a lot of garbage in the way woodwind vibrato—specifically flute and double reed vibrato—is taught.
Flutists, oboists, and bassoonists use the same basic physiological mechanism to produce vibrato. I often read or hear debates over what, exactly, this mechanism is, with some arguing fervently that it is the “diaphragm,” and others insisting that it is the “throat.”
It’s worth pointing out here that a major issue in wind-instrument pedagogy is the fact that so many of the important techniques happen somewhere inside the body where they cannot be easily observed. (Violinists don’t seem to have much disagreement about what part of the body to use for vibrato.)
My belief is that neither the “diaphragm” nor the “throat” can be correctly identified as the organ of vibrato.