Also, you would need the fridge to be incredibly cold to actually cook on it. Cold enough that it wouldn’t really be useful for refrigerating food. But it would be interesting to have a stove that heats food by simply relocating heat instead of actually generating it.

Refridgeration setup:
Compressor -> Condenser -> Expansion -> Evaporator -> start

Ok, there’s maths involved here, but I’ll lay it down real simple: The difference in extremes of temperature between the condenser and evaporator are controlled by two factors: the gas used and the amount of compression used.

No gas is available to make that plate get any hotter than about 150 degrees; At least, not without making the fridge into a freezer. Meanwhile, the temperature of air blowing into the inside of the unit would vary wildly from slightly above near-freezing, all the way down to freezing alcohol. In other words, using it as a controlled heat source renders it useless as an ice box.

Meanwhile, there’s no reason you couldn’t use the condensation end as a pre-heat feed into your water heater (or, in addition to an ice/cold water dispenser on the fridge’s front, have a hot water dispenser) making use of the waste heat as needed, and leaving control in the hands of the fridge’s internal needs.