The problem with the amplifier doppler shift theory is that it hinges on a redefinition of the term "doppler shift." At least in science, terms have precise meanings, almost always defined mathematically. Redefining the terms renders the terms meaningless.
The doppler shift refers to a shift in the frequency of a wave due to relative motion between the receiver (the person or the recoding medium in this case) and the source of the wave. The electrical signal traveling through the amplifier cannot have a doppler shift with respect to the receiver because the person is not a receiver for the electrical wave. The person only becomes the receiver when the signal is the sound wave. Whatever happens inside the amplifier has nothing to do with doppler shift.
The original proffered argument for doppler shift in audio equipment was that the motion of the microphone transducer or the speaker driver relative to the sound source wave source or sound wave receiver, respectively, caused a doppler shift. This is also untenable because it falls outside the accepted scientific definition for doppler shift. The source of the wave and the reciever of the wave are independent of the existence of the wave in the definition. The motion of the speaker driver creates the sound wave; the motion is the source. Hence, there can be no doppler shift resulting from that motion because the wave does not exist without the motion. The same thing is true on the microphone end. The motion of the transducer creates the electrical wave. So there is no doppler shift resulting from that motion, by definition.
The receiver has to be able to receive the signal whose source is in motion relative to the receiver. If not, there is not a doppler shift.