Of course, the mathematician might claim that the "entities" that they're manipulating are more than just the syntax, and are actually much bigger. That is, they might answer "semantically".

Models are semantics. The whole point of models is to give semantic meaning to syntactic strings.

I haven't studied the proof of the Löwenheim–Skolem theorem, but I would be surprised if it were as trivial as the observation that there are only countably many sentences in ZFC. It's not at all clear to me that you can convert the language in which ZFC is expressed into a model for ZFC in a way that would establish the Löwenheim–Skolem theorem.