Since I can set the speaker's position however I like by setting the
DC bias, it follows that I can change that position while the high
frequency sound is playing back on top of it. I could move it at say,
1 ft per second -- this speed is far too slow to produce a sound, and
is practically DC. If you agree that I can set the position of the
sound source by adjusting the DC bias, then it follows that I can
*change* the position of the sound source by *changing* the DC bias.
Any change in the position of the sound source is movement of the
sound source (by the definitions of "movement" and "position"), and
any movement of the sound source creates doppler distortion. QED.

This is basically correct reasoning.