Are you attached to that fence?
You have a lot of suggestions to work with, but they all sound pretty expensive, time consuming and labor intensive. Given how tight you want these tolerances, the issues with garbage, have you considered replacing the wood with solid PVC panels? It might be the right answer; fast, economical, no gaps (you would have to do something about the bottom since a natural stone wall won't ever be smooth and level across the top) no knots, no staining. Buying boards, cutting and staining, you'd basically be constructing a second fence from scratch, and given what you've said about tucking garbage in gaps I don't know that you'd be 100% happy with the outcome... that's a big investment for an uncertain outcome.
