I recently asked a good friend who is a professional gunsmith about this same problem, and he said he had made several piloted countersinks. Turned it on a lathe and then hand filed the teeth, kind of like the teeth on a ball mold cherry. I haven't gotten around to making one myself, but it sounds like a good idea. Maybe someone on here has made one and has a photo?
If you haven't already, you might want to do a search on "counterbore" and "piloted countersink", to see the difference. The counterbore bores a secondary hole in line with the axis of the first hole. The piloted countersink just chamfers the hole, with the pilot keeping it on center. (Added: Or as Jim K notes below, some are a combination.)