chicken manure is hot, needs to be aged.
I have two coops, and after I keep birds in the second coop, and they decimate the greens, it takes a long time for the plants to come back. Granted, I am not moving the cage every day...but that will be pretty labor intensive too.
There are plenty of accounts on the internet of people who keep chickens in pens, move them each day, and end up with wonderful pastures (not dead grass). For all the ones I can remember, they start with a good amount of plant cover, rather than starting on bare soil. So doing this on bare soil is likely to give quite different results.
Fresh chicken manure does not kill established grass, if there is not too much manure. But larger amounts, or plants that are not well established, can make a big difference. So a few chickens wandering on a lawn or pasture will leave an amount of droppings that helps the grass instead of killing it, but concentrated chickens for even a single day have the possibility of leaving enough to overwhelm a few struggling plants (if they didn't already eat or trample the plants.)
The more chickens in the same space, the higher the chance that they leave too much manure in that space.
The scratching is a good point. Layers may scratch more than the broilers that are more commonly used in such situations.I don't think really that chickens will improve your soil, if the problem is organic matter. They will tear the soil up, eat the roots, remove the plant life there.
Planting a cover crop and then moving the chickens there to kill it & scratch up the soil might also work.I think you might be happier and get better results with planting a cover crop, tilling it in, planting it again, collecting manure plus bedding, putting that on top, and a whole lot of time. We live on pure sand, due to seas thousands of years ago, and that is how I added organic matter to my garden. It is slow.
Rather than collecting & hauling bedding, one can put a "chicken tractor" on a given spot, add bedding every day, and move it once the bedding has reached a good depth. That adds a lot more organic matter to each area, but covers the area much more slowly.
Talking to someone local is always a good idea, because conditions are so different from one place to another!I would talk to your cooperative extension agent in your community, for advice on improving the soil.