I was thinking maybe I need to like grease up the nut. I've heard to use a pencil? Does that really help tuning stability? I'm sure it would at least make the needle less jumpy when I'm tuning.
The strings can bind in the nut, which is how you get that "ping" when tuning. If they are catching a little bit, you'll get that jump you're talking about.
A properly set up guitar shouldn't go out of tune that often. It's before 6am on saturday and I nearly wrote "town" .
I compulsively check my tuning at any break in any song during our set. If I don't have to adjust a tuner, I still took that half second to check tuning anyway - so locking/trem/wraptail becomes irrelevant IMO. As my producer/engineer friend likes to say, "ABC - Always Be Chuning". Having a properly set up guitar just means you'll have to make actual adjustments way less frequently.