If you want to poison the soil with a safe product, get some 20 Mule Team Borax laundry detergent and start shaking it out. Nothing will ever grow again. Plants need a little boron to exist but if you get just a little too much, BAM! It's all over.

The tarp might work. Use the silvered ones as they block more of the light than any of the others.

What I would do is cover it with deep mulch? Mulches do not completely prevent growth but they do make it very easy to pull weeds. The roots cannot get a good grip in the loose mulch. And the kids can play on top of the mulch. Our local playgrounds are covered in mulch.

No, you cannot suppress evverything organically. A tarp over the area may create more problems then it would cure because that tarp would tend to keep the soil wetter than soil exposed to sunlight. Mulches might offer some help although on the track they would hinder the cars more than help. Boric acid might help control some plants but it has no affect on others and needs to be applied in amounts that can be toxic to humans.
My kids, many years ago, simply removed any new "weeds" that grew in that race track each time they went out, and more often than not changed the race course.