We have pineapple weed growing all over our dirt roads here, that might do the trick. It's a close relative of chamomile so that might work for you too, but I don't have any experience with it. Daikon radish are famous for breaking up compacted soil.
Personally what I would do is find somewhere else with extremely compacted soil in your area (dirt road?) and see what plants are growing on it, then collect seeds from them or buy seeds of the same variety. This will give you plants that are well suited to your climate.
Not bur dock; you don't want the burrs when it goes to seed . What you want is curly dock which has seeds more like beats. It penetrates blue clay on my property. Roots have the same usability as burr dock best harvested in winter for the stored nutrition for the next years seed production. Dandelions may show up on their own If it is sandy soil plant a cover crop of wheat in the fall; it has high energy seed that can get started without much soil life. When you harvest the wheat the stubble makes a protective cover to start new seeds which feed on the decaying wheat roots. Feeds my chickens and my soil.