Are sets just predicates with syntactic sugar? Do mathematicians agree/accept that "sets are just predicates with syntactic sugar"?
If not, then Why not? I mean, I can translate between $ x \in S $ and $ S(x) $. Will that change the correctness of a proof?
Is there some nuance in university-level math that makes the distinction necessary?

Set theory is a form of logic, and logic is a form of set theory. In my opinion this is made most clear in the form of category theory.
However, I would not expect the "typical" mathematician to be sufficiently fluent in formal logic to deal with this comfortably.
In my opinion, the typical mathematician works in first-order logic on top of (some fragment of) ZFC. Having two "universes" like that is a subtle and confusing thing — maintaining a sharp distinction between "predicate" and "set" is how the typical mathematician keeps them straight.
