What is this amplifier doing, exactly?

oric_dan:

flows for appreciably less than half of each cycle

So what is the class if current flows for half of each cycle, as here?

As I recall, Class C RF amps rely on the low-duty "ping" to stimulate the LC tank,
and set it to oscillating.

Class AB amps as in HF ham radio linear amplifiers rely on the tank circuit to generate the entire other half of the waveform. Class C amplifiers are non-linear, but that doesn't matter for FM or CW amplification.