To be clear, I don't intend to pick a core mechanic and THEN decide what the game is about. The purpose here is to talk about system in an organized way, which means starting with core mechanic. I am interested in what people find compelling about different forms of core mechanics, for the very purpose of helping me determine what is a good one for the game gestating in my brain.
Not sure if this is what you mean by core mechanic, but I like games where the players (everyone at the table) roll dice only when the outcome is both uncertain and likely to be significant for the N/PCs involved, and for outcomes unpleasant to all concerned to always be possible. So in combat both parties can hit, or both miss, but there is always a cost to it.
I don't like rolling more dice than comfortable for in cupped hands, so 2 to 5 max, 3 best.
I prefer rolls to determine how successful, or how unsuccessful, the action is to be bell curves, but I like the rolls determining the outcomes to be linear. 2 separate rolls, neither more nor less, but ...
I like extreme failure and success rolls to explode in a linear way.
I like levels, skills, etc. to only move the bell curve of success / failure, not change its shape. Not do they change the outcomes - equipment etc. can do that. High levels skills etc. don't roll extreme successes more often, but are more successful on average.
I like opposed rolls, so assign levels to challenges.
I hate grinding - so events involving dice rolling should be short.