Indeed, people underestimate the potential of cold-frames. They function like a mini-greenhouse that you can't walk into! In fact, some people combine the two forms to make what is referred to as a pit-greenhouse! I created another plan/drawing of one for you. Picture digging out a 2' wide trench down the middle of the cold-frame floor, deep enough to walk thru, and a "door/hatch' at one end instead of along the length. See below and you'll get the idea.
Indeed, the smaller the greenhouse (or cold frame) the more you will need to keep an eye on conditions and keep them from fluctuating too far beyond the needs of the plants. But if you take the time to set it up well, and think things thru with a good amount of common sense, you should be able to create something that will work well for years to come.
I have seen a few Pit Greenhouses, and they are actually very cool! (ie: interesting designs... not necessarily cool temperature-wise!) However, speaking of that, having them underground does indeed help to cool and stabilize the inside environment! (As you thought, yes it does!) It may not do miracles, however it does indeed help to bring you closer to the "zone" you are shooting for.
(Greenhouses simply recreate different zones, depending on how we manipulate their inner environment and workings.... temperature, moisture, humidity, heat, cooling, etc. by how we implement and use things like shading, heater, A/C or swamp cooler, pumps, humidifiers, lights, etc.)
If you don't have much to spend, then patience, work, ingenuity* and effort can get you there. If you don't have those, then indeed lots of money works too!
Well again, good luck.
*
Think about it.... even with no electricity, a small solar powered pump and solar powered fan can be used to make a small solar powered "Swamp Cooler'!
(I did one some time back! Small, but it worked!)
Indeed, living in the south with hot temps, modifications to the basic design to keep plants lower, where the cooler air will be, would be possible... but will require stooping down to tend plants. Not a bad thing if it gives you a good place to grow and you don't have bad knees!
Things are only impossible until someone thinks of a way to make them possible! It wasn't that long ago when flying was impossible for man, let alone walking on the moon!!!