If I didn’t mention it already, a long pitchfork of strong build is most practical way to break up and turn over the dirt. You push it in like a shovel then break it upward.
Gtaus, would if be a fair presumption your run isn’t moveable?
I have had excellent results using brooders and small coops to compost exactly how you have; scraps, lawn clippings, wood chips, ect. Then after months in place I move the coop or brooder and make it a garden bed. I’ve never had...