I thought the MOV conducted above its breakdown voltage (generally
around 300V), and the energy in the section of the AC waveform above
that voltage heated up the MOV.

Am I missing something?

Ohm's Law:
If the MOV conducts at a low resistance, the power it dissipates will be
minimal. Thus, the surge energy gets dissipated in whatever ground it's
shunted to.

How low is low?

I suspect that's not a complete explanation, but I won't argue at this time.