Yes. And leather seats will make my car go faster, too.
Kidding, bro.
Usually, pickup rings are 100% cosmetic. no tone difference.
The greatest benefit would come if someone made wood toned mounting rings. You know, the kind of papery stuff you see on cheap furniture and speakers? Then you would have "wood toned" mounting rings. At that point you could say you have more "wood tone" on your guitar. The statement itself, like leather seats and tinted windows, is generally enough to satisfy the desire.
The real trick however, is that cars with wood tone panels, (generally old station wagons a-la Chevy Chase in Vacation) are usually slower. So the question is, would wood toned mounting rings make the guitar slower?
I read somewhere that if a pickup is screwed directly to the body strange and wonderful things happen. I thought maybe I could achieve that with a wooden pup ring.
No, because you're still suspending the pickup with a spring. Doesn't matter what the ring is made out of.
Personally, I don't buy the whole direct-mounted pickup spiel, because a magnetic pickup only detects the vibration of the strings. It's not picking up sound waves or the guitar's natural resonance -- the only reason those things are important are because of their effect on string vibration. Direct mounting has no effect whatsoever on string vibration.
I own a guitar with direct-mounted pickups and I can't tell the difference tonally. The only advantage as far as I can see is the cosmetic benefit of dropping the pickup rings entirely. The disadvantage is that you can't adjust the height. Seems like a lousy trade to me.
If you mount directly to the body, how do you change the height? Some kind of spacers?
On direct mounts, some use foam weatherstripping tape underneath the pickup. You can adjust the coarse height by the amount/thickness of foam you apply, then you can fine tune by how far you tighten the screws down and compress the foam.
I read on harmony central a guy put slim pieces of birch wood between the sides of the pickups and the sides of the holes on his guitar under the mounting rings so the pickups made contact with the body. He said it greatly improved tone and sustain.