I have 9 birds in a 70' x 70' yard. They keep the grass chopped short. A bunch of weeds grow n there that they don't eat, so we mow them, but there is a fair amount of grass as well, that never gets a chance to grow very tall. If the area were smaller, I figure they would pull it out by the roots and kill it, the was they do in standard runs.
One thing you can do is grow it in flat boxes with wire over the top, a few inches from the soil. They can clip the grass without killing all the plants that way. I know people have a few of these and rotate them out of the run so they have time to recover. If your mower snips the grass short, you can give them some of the clippings, but do limit the quantity as too much of that can cause crop problems.