Welcome from Michigan!
I'm with you on not growing corn. All it seems to me is a big sign that attracts every critter around to come stomp through and raid the garden. From deer, to turkeys, coons, why would I want to attract them when I can go buy it locally from someone who's set up to or has enough that they can absorb the losses? I agree get it from the neighbor!
On the subject of livestock and gardening I saw a unique idea (I wish I could remember where I saw it at Id gladly share the link) where you have your coop and you fence in 2 large runs between the coop, one on each side of the coop. If you have a heavy breed of chicken you can just have a high fence or you could clip the flight feathers (which is totally harmless its done in the pet hobby all the time for those who have pet birds) so the chickens couldn't fly. Anyway the areas are fenced off and you have your 2 runs with a coop in the middle. The coop has 2 chicken doors, 1 on each side giving the birds access to each run.
Lets say the runs are on the East and West sides of the coop to make things easier as I explain this.
For example in Year 1 the east run has chickens in it. The east door on the coop is open so chickens they can go in and out, eat, roost at night etc. and the west door is closed so your birds can ONLY access the east run. The west run is tilled up and planted as a garden. Then when the gardens all done in the fall, the east door is closed shutting off access to the east run, and the west door is opened and the chickens have full access to the garden. They eat whats left, scratch, poop etc in that area all through the following year. No need to till it, just let them in and have at it. The chickens continue to have access to the west run all through year 2.
Now on the East run in year 2 door stays closed and now the east run is tilled and planted in the spring. The plants benefit from having all the animals on it from year 1. In the fall when the garden is done, you shut the west door, and open the east door just like you did in year 1 and the cycle repeats itself and the chickens get to stay in the east run all through year 3. In year 3 you till the west run and plant it as a garden.
So each year your chickens benefit from having a bunch of fresh greens to eat, and a new place to explore, chase bugs in, and be chickens in. The ground gets the benefit of being rested for a year by not being planted and it gets replenished by having the chickens on it for a whole year.
Another thought would be to go with tractors. Do a search for chicken tractors TON of info on those out there. You can easily tractor rabbits the same way but you have to put something down like some strips of metal like aluminum or threaded rods to keep them from digging out. I'm planning on doing this when I get my meat rabbit project going. Great way to finish weaned ones so I'm cutting feed costs and raising healthier meat. I've even toyed with the idea of putting the rabbit tractors in the garden (some folks do this with chicken tractors too) between the planted rows so they deposit their manure right into the garden. I just hate to waste those little time released bunny nuggets of gardening magic (aka rabbit poop)

Another idea I saw was someone built their rabbit hutches OVER compost piles so the manure went right into the pile. That was for their breeders and they put their weaned rabbits in a tractor and raised them there till they got to processing age/weight. LOTS of really unique ideas out there that you can apply or modify and put to good use..
Sorry to on like that but its something I'm pretty excited about and am DYING to get these projects started.