Could a broken underground wire cause enough heat to melt the snow
At that spot and not kick out the circuit breaker?

Absolutely. At the break, current is still flowing, most likely to ground (earth). The impedance (AC resistance) is so great due to a dirty break and impurites in the earth, the current is high enough to create the heat, but low enough that the breaker does not trip.

The other problem (other than safety) is that whatever you are feeding on this circuit will be running at a reduced voltage. This is bad for many electronic devices.

Just to state the obvious - don't dig without turning the breaker off first!

Since it is creating heat, it is drawing current. Remove the loads that are fed underground, and clip amprobe around each conductor one by one to see damaged wire, It more than likely damaged other wiring.