You have been focusing on the easy case, where Alice and Bob both measure spin in the same direction, so there is perfect anti-correlation and a "Bertlmann's socks" type of argument, which is the argument you are making, is workable.
However, such an argument is not workable for the hard cases, where Alice and Bob measure spin in different directions, at angles for which the correlations violate the Bell inequalities. For those cases, I don't think you can claim that "it's all just due to the previously prepared state", since the whole point of the Bell inequalities being violated is that there is no possible "previously prepared state" (no set of local hidden variables) that can account for the correlations. The QM "state" can do it, but only by being nonlocal, i.e., giving correlation probabilities that do not factorize as described in Bell's paper.