Bells inequality is a statement about math, it gives an inequality that could only be violated if there were no local hidden variables (read: if measurements were truly random). That was a statement of math, which is rigorously provable. It took experimental confirmation, but we can now say with high confidence that there are no local hidden variables (i.e. there is no information hidden that we simply cannot measure, instead the outcome is only decided the moment you measure).

Global hidden variables are still an option, but they would require much of the rest of physics to be rewritten