Bell's theorem outside of the context of entanglement I am an undergrad college student and have been watching videos on YouTube explaining Bell's theorem / inequality and how it shows that there cannot be local hidden-variable theories for entanglement.
However, the experiments most videos talk about are in the context of entanglement specifically, where the experiment involves taking a pair of entangled particles and recording the statistics of how often the results match when two people repeatedly measure the two particles along randomly chosen axes. The statistics of the data we get after performing the experiment rules out local hidden variables.
My question is how this rules out local hidden variable theories in quantum mechanics outside of the context of entanglement? Or does it actually only rule out local hidden variable theories in the context of entanglement?
For example, we could have a local hidden variable theory outside of entanglement like the following: A single electron (no entanglement) could have a definite position before it is measured (a local hidden variable), but we cannot know it until we have measured it, and that local hidden variable is why we can only talk about the probability of finding it at a location. And therefore, the randomness is not a fundamental property of the quantum nature of the electron but is due to a hidden variable.
It seems to me that the Bell's theorem experiment they talk about in the videos only tells us that quantum entanglement specifically cannot be explained by local hidden-variable theories, but not that other quantum phenomena, such as the fact that we can only know the probability of finding an electron in an atom at a certain location until we measure it, cannot be explained by local hidden-variable theories.
Are there other experiments that have been done that rule out local hidden variable theories in these other contexts? Or if not, how do you generalize the result of Bell's theorem/experiment in the "entanglement case" to prove there cannot be local hidden-variable theories in quantum mechanics in general?
Clarification: I am not for or against using any theory. I just don't see how local hidden variable theories in QM in general, are ruled out by the Bell experiments with entanglement specifically and am asking how/if local hidden variable theories are ruled out in general.
 A: 
My question is how does this rule out local hidden variable theories in quantum mechanics outside of the context of entanglement?

It doesn't! That is a good point, Bell's theorem applies exactly to the system that Bell described and it does not globally rule out the existence of hidden variables. And, though there are many other systems for which you can come up with similar inequalities, there is no such inequality for single-particle systems. And it is for that reason that the logical possibility of hidden variables for single-particle systems is not excluded by Bell's theorem, as you suggested. So, there could be hidden variables under the hood, but if they are local and realistic (so the mainstream interpretation of Bell's assumptions goes), then those hidden variables may explain single-particle phenomena, but not the spooky results we get from entanglement.
A few things to say about that:

*

*No matter what, you can say that the universe does have phenomena which are not consistent with the classical idea of a local, realistic hidden variable theory (LRHVT). This on its own makes Bell's theorem interesting and shows that there's more to the universe than what people pictured before QM came about.

*Although you could hypothetically explain single-particle phenomena with some LRHVT, the reason that Bell asked himself "Could we have a LRHVT?" was to know if it would be possible in principle to find some kind of very down-to-earth explanation for entanglement, if we understood it better. And it turned out this was not possible. In contrast, the results of single-particle measurements don't beg as much for an explanation, because there is no apparent instantaneous effect over large distances, which Einstein originally claimed was in contradiction with relativity.

