OK so I am feeling helpful today and I'm bored. I also enjoy playing in CAD so here is a drawing illustrating a basic design for what you are after using stock scaffold frames and standard sizes of lumber and plywood. This is a 45' diameter platform which uses 14 x 54" x 7' scaffold sections and 5 x 54" x 10' scaffold sections. Those are readily available rental scaffold sizes. Each frameset would need screwjacks and foot pads of 2x12 to prevent sinking unevenly into the playa. On top of each set of frames, you would place two 4 x 6 beams in either 8' or 11' (could be 12') lengths. The red lines represent 2 x 6 stringers on 4' centers with no span reaching in excess of 8' between a structural member. Between those you would use standard framing joist hangers to install 2x6 spreaders on 2' centers (not shown). Those could all be precut before you leave for the playa as they would all be the same size. You will need to do some custom framing around the edges due to the curve but the bulk of this design is easily predone using lumber in lengths you can buy off the shelf. Assuming you would likely end up with a large crowd on top of this huge stage, you need to build it like a house.
The rectangles represent 4' x 8' sheets of 3/4" ply. You could double surface it if you wish in which case you would run the second surface at 90 degrees to the first. My recommendation would be to build the platform in place and use full sheets of ply and let them hang over the edge, then take a heavy duty router on a long bar with a center pivot and scribe the circle out on the playa. You could precut the circle, but getting all that plywood to line up again a second time would be a pain. You may need an extra leg or two around the outer edges depending on your use and what sort of fascia you have in mind. The fascia could provide all the secondary edge support you need.
When you are done with the project, except for the odd cut pieces around the edges, you will be left with a whole lot of uncut lumber in very reusable lengths and sizes. Or burn it... up to you.
Have no doubt, this is a shitload of materials and will take a 6 - 8 man crew about two days to construct before you even think about putting something on top of it. Enjoy and good luck!
A plan is what you vary from.
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