[Physics] Does this Bell’s experiment actually disprove Local Hidden Variable Theories (LHVT)

bells-inequalitylocalityquantum mechanics

I'm watching some archived video lectures on QM in Coursera given by Umesh Vazirani from UC Berkeley and I have a question regarding a Bell's experiment (I guess something close to this) described in the lecture. I'm going to link to some of the videos, hoping that I'm not infringing any copyrights. If you think I am, kindly let me know and I'll remove the links immediately.

I don't know if it's an oft-used one, but the experiment is described as such: There are two boxes far apart with inputs 0 and 1 for which both of these boxes should output either 0 or 1 so, that if the inputs are both 1 then the output bits should be different, otherwise the output bits should be the same (as in same bits for both boxes, it doesn't matter if the bits are both 0s or 1s).

Now it's said that if it were for some hidden variable, the highest rate of success for this experiment could be 75% (both boxes always outputting 0 or something like that). This is the claim I can't agree with, but I'll get there in a bit. The second thing that's said, is that if we use QM then the success rate can be higher (85% is mentioned which is roughly equal to $cos^{2}(\frac{\pi}{4})$). The video overview of this experiment can be found here.

There's also another video which explains how the 85% success rate is found. Although, my own calculations show that the maximal possible achievable success rate is approximately 89.5%. This can be achieved by slightly adjusting the $\theta$ used in defining the measuring basis. But this is besides the point.

Now my question is: Does such an experiment really disprove all the local hidden variable theories, or is it simply Wittgenstein's ladder in action? I think the local hidden variable theories weren't given a fair chance. Would the success rate be lower than the 89.5% I mentioned, when we "entangle" the particles as we did in the QM model and run the same measurements, but expect no actual entanglement between these particles. Which means that instead of entanglement we describe these two particles as synced up so that their local hidden variables will cause the measurement readings to yield certain values. This way we can use the knowledge that they're synced up to build two boxes which might have the success rate as high as in the QM model.

Now reading up on Bell's theorem from Wikipedia I'm given the understanding that in an actual experiment the success rates might differ, but ever so slightly. That difference, I understand, comes from some kind of a Bell's inequality defined by the actual measurements done on the system. Is this more or less correct?