What purpose does the mounting ring have? I don't really understand why they're there. Can you take them off/will they work if the mounting ring is removed?
it hold the pickup in place, and lets you adjust the height of it. otherwise it's just laying free in the cavity.
If you take the mounting ring off of a pickup, it'll just fall out because the pickups are mounted on them. And if you screw the pickups directly into the body, it will probably be ugly because that part of the guitar isn't meant to be seen.