What I am wondering is, where does this kind of consideration break with traditional computability theory? Is traditional computability theory limited to what Turing machines can do, while perhaps it is straightforward to prove that the operation of this Universe requires computation beyond what Turing machines can do?

There's a large set of computability models, but if you don't get into hypercomputation they all produce the same set of computable functions. Quantum computation doesn't change this picture; anything computable by a quantum algorithm is computable by a classical algorithm, although often less efficiently.

Whether or not the physical laws of the universe involve any uncomputable operations is an open question, although none, as far as I know, have been proven to exist.