Clockworks always play out the same way and so we can predict the position of the clock at some future time with precision.

Are you sure about that? QM and GR might disagree.

I guess my question is how do you prove that no hidden variables exist?

Hidden variables has been a contentious point with QM since Einstein and the EPR paradox. But, the worm has turned, as they say, and the present consensus is hidden variables do not exist. The Bell Inequities, Alain Aspect's and Nicolas Gisin's experiments show that hidden variables has some fatal flaws that preclude their reality.

One does not prove that hidden variables do not or cannot exist. One can only show the preponderance of the evidence is against such things.

IMO one of the best, comprehensive and lay-friendly write ups on whole controversy is

here