I propose a convention for Checks vs what I want to call Tests. Checks are what we're used to, roll a d20 (or funky die?) and add your mod. While Tests would be pure TESTS of Strength, Luck or whatever. No classes or occupations would be in consideration. I think this is very much a Save in C&C? (I don't know, I don't own it.)
The concept of the "save" in C&C is still a d20+relevant bonus/penalty -- in C&C, you have a specific DC built in (typically) and it depends on your "primary" ability, which is chosen at character creation. So, for example, if your DEX is 15 and it's a "primary" the base DC for a DEX save is 12. If DEX isn't a primary, the base DC is 15 (I think...yeah...15). The base DC can be adjusted up or down by a Challenge Level. So, if a lock is deemed CL 4, a person with DEX as a primary would have a DC of 16 to pick the lock, while someone without DEX as a primary would have a DC of 19 to pick the lock... So, you still roll over... is the short answer.
AD&D was the "roll under your stat" mechanic for skills and such.
I'm all for a unified system. For Luck, maybe make a "luck check" of a certain DC and if you roll over it gives you a bonus to the actual roll? Dunno. We've been using Luck as simply a burn mechanic (like action points in 3.5)-- not as an actual skill roll. I haven't completely digested the core rules -- waiting on the physical book to save my eyes!