They do eat it. If it starts out too tall, sometimes they wont eat it because it is tough I think. If it is shorter to start out with they usually eat it down to dirt! If your flock is too small to eat it all down, mowing would be more for aesthetics than any other reason.
I put 6 pullets in a run that is roughly 200 square feet with grass that had not been cut in months, it was about 7-8 inches high. After only a month you would have been hard pressed to even find so much as a blade sticking through the fence from the outside.
They are rather rough on the turf. I think most of the destruction of the turf comes from the scratching.
Some people have gardeners they hire.... I have moonscapers that mostly free load off of me.