Hi Joe, welcome to Permies!
I like your idea, and I will disclose that I am not an engineer. Where abouts are you located? Are in you an area with any seismic risks? I think the dry stack stone will work. The footing however, I believe could be challenging to get it right to avoid possible future problems. You mentioned clay soil, and clay soils are expansive, some more than others. It's this expansion and contraction that can cause minor to major problems in conventional homes such as cracked foundations, cracks in brick work (like the stair stepping cracks going up walls and corners), cracks in interior sheetrock walls, doors that once closed no longer close, etc. All of those are symptoms of foundation movement, either from a poorly constructed foundations, expansive soils, or freezing and thawing. I have a few ideas that may help you in your quest.
-Check out the USDA web soil survey found here:
https://websoilsurvey.sc.egov.usda.gov/App/HomePage.htm In it, you can zoom in the map, click the AOI button to draw an Area of Interest around your building site, and then click the tabs across the top to view soil information. In it, you'll find oodles of information, like the name of the soil, expansive qualities, how well it drains, thicknesses of the horizons (layers), depths to obstructions (like bedrock), and more. This will have information on how little or bad the soils expansiveness.
-Expansive soils can be managed by backfilling a foundation with crushed limestone, like #57 for example which has the fines removed. This type of gravel readily moves with each stone slipping by its neighbor and will act as a shock absorber, allowing the soil to move and help reduce or possibly eliminate lateral compression stresses on the foundation wall. The worst of the highly expansive soils can exert a compressive force of a million pounds per square foot. It may only push a fraction of an inch, but they can push hard.
-Gently slope the footing trench so water all runs one direction to some exit.
-Fill your footing trench with river rock 6 inches at a time, and run a plate compactor over each layer.
-Try to incorporate some sort of french drain around the footing.
-Slope the finished soil grade away from the home in all directions to help reduce the amount of rain that infiltrates the soil around the house and heads down towards the foundation and footing.
-And lastly, consider installing jacks between the sill plate and your subfloor. This can make it relatively easy to lift and re-level portions (or the entire dwelling) if the foundation settles in areas.
I'm not familiar with the use of a non-monolithic rigid footing to build on, but that doesn't mean it can't be done. Others may have
experience in this area and have better advice. Hope this helps!