I use a 3 1/2" hole saw with a 1/4" pilot on a drill press stand to cut out the plywood disks. The plastic/vinyl nipple pad is cut with the same hole saw with a hand held drill motor on some scrap Homasote.
I use a vac on the drill stand to keep the circular cut clean.
You can see the 3 1/2" scars from cutting the disks on the scrap 2 X 12.
I then insert a large landscape timber nail into the pilot hole and hold the disk with friction on a router table with a 1/4" round over bit. The disk rartles around somewhat and is a tad nasty but it gets the job done.