I think that depends entirely on how the run is managed. IMO, any bare soil is unhealthy soil. So, if you are able to keep a healthy cover crop or mulch in place, the soil WILL improve. And not just a little bit, but it will improve dramatically. How often have you driven by a chicken run and seen the birds lined up at the fence, staring at the grass growing just outside beak range on the other side of the fence, while they are left in a barren moon scape that is alternately a dust bowl or a mud pit, depending on the season? I see it way too often to keep count. But, if you take that same barren run, and put a nice deep layer of mulch (6" is just a beginning, because that mulch will quickly melt into the soil as all of the beneficial organisms go to work on it) over the run, the chickens will have a never ending source of entertainment and food. They will flip through the mulch, harvesting the insects/worms/and even eating a fair share of the mulch.I don't understand that. That's the complete opposite of everything I've seen/read.
Vermont Compost company uses chickens as an integral part of their compost manufacture operation. This business model is a winning proposition for business owner, local towns and businesses who are able to divert their waste away from local land fills, chickens who derive all of their sustenance from this operation, while providing free eggs and enriching the compost with manure, and the many customers who purchase the finished product to enrich their gardens.
John, there are many directions you could head with your garden/poultry operation. You could even divide your garden into 4 blocks, with the coop in the center. This would give you more options for management than the 2 block style. How big a garden do you REALLY need? Are you growing more than you need right now? Is a lot of it going to waste? Do you anticipate adding any other type of lifestock in the future? Are you working harder than you need to to grow what you want to grow? Are you planning to donate/sell excess veggies? Will you cover crop? Do you do a lot of tilling throughout the garden season? What's your climate, water availability, soil type? I'm excited about the many options available to you and hope that you will supply us with photos and an ongoing write up of how this plan unfolds over the next year or two. I'd also be interested in a before/after soil analysis.
Reading back through your post, I came to the section where you are hoping for minimal feed inputs. While your birds will eat a lot of the greenery in your garden spots, I doubt that the plant material will be varied enough that they will have their nutritional needs met. I expect that you will need to feed them on a daily basis. But, you could grow some cover crops that are protein dense, and also grow some crops that bear a fair amount of seeds that they could harvest directly in the garden. I don't think you'll be able to expect them to have the soil weed free and seed ready at planting time. But, they will do a lot of pest management for you!