I have lush green grass everywhere in my back garden because mine are free all day and fertilise it with their droppings.
They dont have a run but have a house which they gladly go to every night to sleep in. I dont need to mow the lawn cause the girls do it for me.
Im lucky here to not have any predators, however if i did have preds, then of course i would have a run and be faced with the same dilemna which is where the mesh laid on the ground comes in handy.
Ive seen it done and it does work, though its best to get really tough rooted grass, the grass grows through the mesh with the roots safely under the ground under the wire, which even the toughest chook cannot budge! ( small grade mesh, not the large)
I wish you the best of luck, be positive!