Everyone would need to know what the frost point is on the ground where you live. The idea is to bury 100 to 200 feet of 1/2 or more black poly pipe 2 to 4 feet depth in the ground beneath the frost point . The ground temp should be 55F you need the hydroponic reservoirs at 65 to 68 degrees F It is the best to dig a trench and bury the poly pipe. Now you must decide how you plan on exchanging the heat ( heat exchanger or heat exchange. Like coiling up some of the poly pipe inside of the RDWC or DWC or Aeroponic reservoir. I don't passive is the correct word because you will need to have a fountain pump somewhere in the system. I would suggest a lower GPH pump for the best thermal exchange. Everyone is welcome to chime in and share your knowledge on this subject
Probably will work. Back in the 70s people in the desert were building homes with cooling tubes buried in the ground and powered by a heat chimney driving convection. It worked well. With a closed loop water system like you discussed might need a variable pump or a regulating valve to control flow. By controlling the flow you might be able to adjust the cooling performance of the system. Maybe. If I needed a cooler for my hydro I would be giving it a try.
I just use my Watercooler if I've to. But my Ebb and Flo Reservoir is about 90 Gal. That Juice stays cool even with no Watercooler.