RP wrote:

It won't even cycle on in sub-freezing weather...

It might, with an old refrigerator thermostat in the freezer and
a Thermocube in the fridge that light a 100 W bulb in the fridge
if the freezer rises to 20 F... or the fridge falls to 35 (to avoid
freezing the fridge contents.) Will this hurt the compressor?

That's pretty neat, but it won't help out with the subfreezing condenser
air. Somehow I doubt that a fan cycle control is a viable option

So subfreezing condenser air might hurt the compressor, and controlling
the fan duty cycle to keep the hot coil above freezing might avoid that?

It might, that is, if you could insulate the coil from the ambient
environment during the off cycle.

With little mass and no fan, it might heat up fast when the compressor starts.

It's doable, but it would be an engineering nightmare on a stock domestic
refrigerator...

Seems easy enough to put a thermostat near the hot coil that turns on
the fan when the air temp rises to 70 F. Or another Thermocube and bulb
or heat tape near the hot coil.

Refrigerators take awhile to stabilize running pressures, so starting with a
subfreezing condenser coil might mean that the cycle finishes before the
head control even has a chance to takes control...

Is that bad, or just a transient self-correcting condition?

Without head pressure control, when the condenser gets too cold the head
pressure will drop to such a level that the evaporator will be starved of
refrigerant, even with a TXV in place, and as a result and superheat will
escalate. The evaporator and compressor are starved of refrigerant. Turtle
had it exactly wrong; flooding back is assured not to happen.

Good.

Nick