Almost all ET PAs are used in Class AB mode. And while the AM:PM distortion can't directly be controlled by the supply voltage, our experience with the vast majority of handset PAs is that correcting the AM:AM using IsoGain ET also brings the phase response into line. And simple memoryless correction of AM:PM ("DPD-Lite") is an increasingly common capability of the latest handset chipsets.
I agree that an envelope-tracking amplifier does not have to be a "class C" amplifier, although that would provide the greatest efficiency, it could easily cause some distortion. The other comment is certainly valid, which is that it takes a great deal more effort to design an amplifier that will deliver a satisfactory level of distortion at various operating voltages. Phase modulation certainly does happen and it must be compensated for in order for the distortion level to be acceptable, and designing the correct compensation will certainly add a lot to the design effort.
The surface response of an ET PA is more complex than a simple 3D RF-input vs battery-supply. Phase pushing and pulling will occur in such a 3D ET control solution.
One comment on the transistion region to be noted; it is the same break-point as system modulation pre-distortion uses. There are two other break-points needed to arrive at the most effective ET modulation efficiency which are not mentioned in the text.
The objection of the envelope tracking transmitters is to improve efficiency of the output signal.
It is collecting the input signal data and replaces it with something that we do not know at the time.
Envelope tracking reminds me of the two modulated stages in an AM transmitter with class "C" driver and output stages. Much more efficient than anything linear. But more demanding on the output filtering section.