Cleaning the holes out by drilling a larger hole and then filling with new wood is the standard way to do this. I'd drill a hole at least twice the diameter of the screw - so you will be tapping into wood rather than a combination of wood and glue. Our local hardware store has hardwood dowels in 1/8, 3/16, 1/4" and up. Use a small sanding block to make sure the ends of the dowels don't stick up above the surface.
You can check with the various luthier message boards, but my guess is that this is exactly what would be done to a vintage Les Paul. If the guitar had a lot of historic value they might try use one of the various super glue versions to harden the threads that are still there. But if the old wood has been stripped out there is no cure but to put new wood back in. The key is doing a clean neat job (even if it is going to be hidden under the bridge).
By the way, using a toothpick is actually another way of doing this same thing. If the hole is small and the toothpick is hard it is a perfectly good repair. Its not quite as good a drilling out and putting in a dowel larger than the screw since the screw will be biting into a combination of the wood of the toothpick and the guitar and the glue. But it is often an effective repair, it's easy, and it doesn't have any chance of doing damage.