I would never discount the advise Alex gives, he knows what it takes to shoot one-hole groups day in and day out. Having said that, I will answer your question directly as I too like my brass shiny and bright, AND case necks that are carbon free. I figure if new off the shelf match ready ammo doesn't have dirty necks, it can't be too wrong to do the same in hand loading. And all my guns shoot 1/4 to 1/3 moa, not good enough to compete, but good enough for me, with the occasional group in the zeros being an excuse to post another target on the walls of my loading room.
I SS tumble in soap and Lemi Shine after annealing, rinse and blow dry using compressed air. To remove any trace of water, I put the freshly cleaned cases in a small cardboard box with two, 2 inch hole in opposite sides. I stuff a hair dryer in one hole and run it on low for 5 minutes. That'll get the brass up to about 170 degrees F and that'll take care of any residual moisture. Never block the exit hole in the box, you'll melt the hair dryer.
Another idea, and I will say that I have not tried this yet, is to add a little dishwasher "rinse aid" liquid such as Jet Dry or Cascade Platinum to your rinse water to eliminate water spots.
In my hometown, the tap water is not particularly hard, or soft, so blow drying works best and cheapest. I intend to try the rinse aid in the near future, just for gits and shiggles.
As for peening case mouths, reduce the number of cases in the batch, and increase media. Cases should not be impacting cases during tumble. They should be cushioned by the media and what's happening is the media is moving into the cases, reducing the amount needed to cushion the tumble.
I hope this helps.