But I admit I'm stuck. What I can't get clear is what exactly the logical influence of the quantum theory is - I can see that the theory predicts that the negatively measured electron spin will correspond to what it would have been if positively measured, rather than what one would expect from classical analysis of an unmeasured spin.

What I can't get clear is what happens to the classical analysis - exactly - when the electron is unmeasured and in all spins.