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Elements of class G/H then?
Not exactly, because it wouldn't use the input signal to control the rail voltage. If the volume knob controls the rail voltage, you don't need to apply any attenuation in the digital domain, so you don't lose resolution.
I was looking at some Class D modules which will run on anything from 12V to 56V, the lower limit presumably being the minimum required to operate the logic chips. If you separate the logic power supply from the HV rails powering the output MOSFETs, you can just use a normal small low power computer PSU to power the logic and you can set the output stage rail voltage to a more or less arbitrarily low value to make it quiet.
Elements of class G/H then?
For beginners, this is a good read https://www.soundonsound.com/techniques/what-class-d-amplification