From Wiki, my bold:
Bell's Theorem: If the quantum mechanical formalism is correct, then the system consisting of a pair of entangled electrons cannot satisfy the principle of local realism.
Although the current Wikipedia article does indeed say that, that is not what Bell's theorem actually says.
What Bell's theorem actually says is that under certain specific assumptions, all of which are assumed by classical physics, certain inequalities must hold.
Bell's theorem is true independently of the experimental evidence.
When combined with the experimental evidence that says the inequalities do not hold, the
conjunction of the experimental evidence with Bell's theorem
implies that at least one of the classical assumptions does
not hold (from which it follows that no explanation based on classical physics can be consistent with the experimental evidence).
I believe the sentence you quoted was intended to explain the consequences of Bell's theorems in light of the experimental evidence. It does not give a correct statement of what Bell's theorem actually says, because Bell's theorem does not refer to the experimental evidence, and the conclusion you bolded cannot be derived without using experimental evidence.
(Even then, there are a couple of other assumptions, such as counterfactual definiteness, that could be rejected instead of or in addition to local realism. Selecting local realism as the assumption that must be rejected is a choice based upon considerations that lie outside of Bell's theorem and the experimental evidence, and that choice is actually mildly controversial.)
The importance of those details to this discussion is that
markie has been saying Mills has given a classical explanation of the experimental evidence that show Bell's inequalities do not hold.
markie's claim is false, because classical physics assumes all of the hypotheses of Bell's theorem. To come up with an explanation that is consistent with the experimental evidence, at least one of those hypotheses must be taken as false.