The problem lay with thermodynamics. To chill the water, you have to remove the heat. Where does the heat go? Into the insulated fridge, which traps heat, but it is still trying to cool the coil, which releases more heat... well, you get the idea.
Additionally, it causes the compressor to run continuously for cooling, which shortens the life of the compressor and raises electric bills.
They really are not made to handle temperature swings, but instead are insulated to keep the inside of the fridge cold with an occasional small heat load. That's why it may take 24 hours or so just to make your case of beer/pop cold.
The removal of heat per hour is called BTU. A dorm fridge may remove 100- 800 BTU on average, and air conditioners may remove 5000+ BTU.
Evaporation is roughly 8000+ BTU per gallon making that the most efficient. I may check out DIY auto top offs.
Now, after having said all that, I have thought about just placing a whole tank in a glass door fridge.