Though it may sound superficially plausible that any particular such random outcome is actually the deterministic output of a hidden pseudorandom number generator, that hypothesis is ruled out by Bell's theorem.
As far as I understand Bell's theorem only rules out the hypothesis that the random outcomes are the deterministic output of a hidden pseudorandom number generator that obeys locality. There could be a deterministic, non local process that generates the "quantum randomness", and this could be detectable if it exists.
Any true non-local effect would however clash with the observation that the universe obeys the theory of relativity. Quantum mechanics can handle this by switching to quantum field theory, but non-local theories such an Bohmian mechanics cannot.
As far as I understand Bell's theorem only rules out the hypothesis that the random outcomes are the deterministic output of a hidden pseudorandom number generator that obeys locality. There could be a deterministic, non local process that generates the "quantum randomness", and this could be detectable if it exists.