I live in AZ also and have indeed used the ground to cool the
nutrient solution. First, the ground temp wont be that cool. Your looking at about 72 degrees 3 feet deep, and it may be cooler deeper (say 10 feet deep or more) depending on moisture level. Five gallon buckets are only about 18 inches tall. So they wouldn't be more than 16 inches deep (leaving the top 2 inches above ground) unless you build a lip around them to keep the dirt out.
Second your planning to grow both large and small plants, and I don't know how you plan to work that out. Five gallon buckets for large plants like tomatoes and cucumbers is too small for me to consider using, and way to large for lettuce and spinach.
It sounds like your planing to dig 20-30 holes for 5 gal buckets (with planing 8 different plant types). There are three issues I see going that route.
First is each 5 gallon bucket will really only have about 3 gallons of water in it at most. And not being very deep the water temp will wind up too high (especially such a small water volume).
Second is each bucket will need an ample air supply for the roots. This poses two problems. One, the cost of buying one or multiple pumps able to supply enough air to them all. Two, pumping hot air into your buckets will defeat the purpose of trying to bury them in the ground in the first place.
Third is maintenance. Small lettuce plants wont be too much of a problem because their easily removed with the lid, and don't drink nearly as much water as the other plants. The larger plants will drink in excess of 1-2 gallons daily, and with only about 3 gallons of water in each bucket, your going to need to do constant maintenance in both relapsing the used water, as well controlling the fluctuating nutrient concentrations from the constant changing water volume in the buckets. Not to mention how your going to remove the lid and root-ball to clean out the buckets and nutrient changes periodically while the plants are growing.
I would say your much better off going a different route, using a flood and drain, and/or
drip system/s. Using large reservoirs at least 3 feet deep. insulating any part of the system above ground.
P.S.
Lettuce will just bolt in air temps above 85-90 degrees (even under shade cloth). That is elongate and produce seed pods. You'll be better off trying to do lettuce indoors in the air conditioning and under florescent
lights during summer in Arizona. Then a simple water culture system would work fine. But I still wouldn't use five gallon buckets for lettuce. Their just the wrong configuration (shape) to maximize space and yield.