One of the best solitions is "physical control" poking holes in the lawn through core aerations, or through lots of holes with the garden fork like I did int he video. Remmeber the fungus isn't killing the lawn, the lack of water permeating the area is.
There are 3 different types of fairy rings, one of which has NO mushrooms and has only a ring of darker grass. They can be relatively small or like one in my yard that has expanded to about 40 feet.