> Cantor's original proof requires the list to be provided in advance
No, it does not. It's an existence proof showing that:
IF a list of reals L(x) (where L : N -> R) exists, THEN a real A_L exists such that for every natural number n, L(n) ? A_L.
Or, equivalently, showing that:
For every function L from N to R, there exists a real r not in the image of L.
Being a constructive proof, it's usually *presented* as if it were an algorithm or a procedure, but it's only for ease of comprehension. Unfortunately, that seems to have backfired on you.
Incidentally, "countable" and "listable" mean the same thing in conventional set theories: an infinite set S is countable if and only if there exists an surjective function f from N to S. Since a "list" of elements of S is most easily formalized as a surjective function from N to S, denying that S is listable is equivalent to denying that it is countable.