Theorems are the highest standard of proof we have, so it's a bit strange to describe a result as "only a theorem".
But yes, the theorem does assume the postulates of quantum mechanics as preconditions. It's not possible to make any statement about reality that doesn't have this kind of "if it works this way" postulate.
yes, i regret using "only"; i meant "only" that theorems are still subject to experimental verification and also a single theorem can support multiple interpretations.
This isn't a very fair assessment. All evidence we've gathered so far seems to indicate that they do. Of course you're technically right to say that "they may not", but that's true of any physical theory, so it's not particularly meaningful.
But yes, the theorem does assume the postulates of quantum mechanics as preconditions. It's not possible to make any statement about reality that doesn't have this kind of "if it works this way" postulate.