I think there are two simple refutations, one theoretical, the other practical.

Theoretical -- If the output of a driver accurately follows its input waveform,
how can there be Doppler distortion?

Practical -- It should be trivial to compute the sidebands produced by combining
(say) a 60Hz signal and 6000 Hz signal. By looking at the phase of the
sidebands, it should be possible to determine what part of them is IM distortion
(AM components) and what part is Doppler (FM components).