Good detailed description of the problem - thank you!

So, yes, coated strings are quite a bit slipperier than non-coated strings. That may be the culprit. It sounds like you are on the path to getting the skinnier strings to cooperate with your fingertips. So hopefully if you swapped out all the "wound" strings with non-coated, you should be able to get better grip.

Thinking about your description, I also wonder if you may be actually pressing harder than you need to, to get the tone to ring out from the note. Experiment with lightening up your touch just a bit and see if that makes you less likely to roll off one side or the other. Just a thought.

Personally I'd steer away from any kind of fingertip goop because it would end up accumulating on the strings and dulling their sound.

Hope this helps! - Lisa