problem is that I have no idea where class C amps have gone
Class C amplifiers very much exist, but they are not used in audio because they inherently have very high distortion. They are commonly used in RF transmitters, with their outputs fed into tuned circuits (tuned to the frequency being transmitted) which can partially filter out the distortion.
The letter D used to designate this amplifier class is simply the next letter after C, and does not stand for digital
Quoting from a post I made in a recent thread on Class D amplifiers:
Whether or not a class D amp is referred to as digital or analog is not related to its class D power output stage. What would properly be called a digital class D amp is one that can accept a digital input, and which puts that digital signal through some dsp (digital signal processing) to generate the modulation waveform that ultimately controls the switching of the class D power output stage. An analog class D amp would differ in that it would utilize an analog input and have an analog front-end signal path.
Regards,
-- Al