If this has been a long lasting problem for you that might work. On the other hand if it is an initial occurrence wait and see if it depletes the organic material it is feeding on and disappears on its own accord. Why do all that work when you don't have to.

Sounds a bit drastic! Does just mowing off the toadstools not work?

Depends which grade of fairy ring you have - grade 3 is just a circle of toadstools which gets wider every year - grade 2 is a ring of toadstools with darker green grass at the edge of the ring, and grade 1 is the worst- you see two rings, one round the outside with toadstools, then a bare or dead or moss ridden inner circle, with a dark green ring inside that. This latter is a big problem, and is the only one that's worth trying to eradicate, though its next to impossible to do so. Trying to eradicate it is done by excavating the turf and topsoil to a depth of a foot, with the excavations extending at least a foot beyond the outer ring of toadstools, with all the earth so removed either being disposed of or reused well away from any lawned area, without spilling any of it, not even a speck, on the remaining lawn when transporting it away. Fill the excavation with new topsoil, level off and returf, or sow seed, and with luck, it should be gone. There is no chemical treatment which will work.