Yep, the cost of the wire really adds up!
They don't really eat much grass, they prefer soft weeds, clover, little bugs, etc. so you probably will still have to mow. And if you are prone to dry weather, I really recommend watering the area they are in or the grass wil become overwhelmed by the nitrogen in the poo and fry. What really makes the clay runs is when you have heavy rain, then the chickens tear everything up then it fully dries out. The ground just can't recover after that happens a few times in a row with no rest. I think it's possible to keep your grass with the right management technique for the grass. Some people put wire over the ground so the chickens can't scratch the crown of the grass up.