I have a run with a net on top and I actually don’t clear out the chickens poop. I dont buy anything to improve the soil either. Just throwing in my home made wood chips from pruning and adding autumn leaves give a wonderful soil...
The "Back to Eden" system that Paul talks about uses his homemade wood chip compost (with animal manures) to get the results he shows. If you just used plain wood chips, it would not work nearly as well. I saw a YouTube video where a guy visited Paul's gardens and said that when Paul talks about this wood chip "Back to Eden" garden, it's not plain wood chips he uses. It's much more of a composted wood chip litter filled with animal manures and other organics. So that is what I have tried to recreate in my chicken run setup, although I use more leaves than wood chips in my current system. It's all organic and breaks down into great compost.
I throw mixed grains in the run for my chickens (almost) every day to scratch and peck a little before they start to roost. The soil is very much alive and healthy. I even never had to deworm my chickens.
I do free range my chickens too. But only part of the day. If I didn’t free range I need more ‘browns’ to compensate the nitrogens in the chicken poop. I have 8 bantams on 15m2 run space. And there are 4 bushes in it. 3 types of berry and a Portuguese laurel.