Of course, all of this is quite familiar to anyone who has ever thought about tests of Bell's inequalities. It's understood that if our measurement efficiencies are low AND there is a systematic effect such that the recorded pairs are a biased sample, then the simple Bell inequalities are no longer relevant. Furthermore, it's obvious that we can imagine realistic mechanisms that yield biases such that the simple QM correlations are reproduced within the biased sample. No such minimum appears in the data. Thus, we must imagine that when two particles emerge from a singlet state in opposite directions and interact with spin-measuring devices, one of them exhibits a definite spin when measured at any angle, but the other sometimes doesn't exhibit spin, if the measurement angle happens to be nearly perpindicular to its spin axis.
Similar comments apply to the polarization of photons.
However, the most compelling factor for many students ends up being the realization that, of all possible dependancies between spin correlations and relative measurement angles, there is a unique dependancy that minimizes the net exhibited non- conservation of spin for all possible interactions, and that happens to be precisely the dependancy predicted by quantum mechanics. Thus, if QM was wrong, it could be shown that the workings of nature are, in a very intuitively palpable sense, arbitrarily defective.