Earlier quoted context omitted.
Physics is based on what can be measured objectively. In other words the probability could be an approximate description of a real (and perhaps knowable) yet un-objectively-measurable process. The idea "everything that is natural can only be everything we can objectively measure" should be problematic right? Of course whatever out there that is unmeasurable is rife with confusion, messiness, lies etc but that shouldn…
You seem to suggest some kind of hidden variable theory for free will: https://en.wikipedia.org/wiki/Hidden-variable_theory . As far as I know they're not particularly popular among physicists, although we haven't (completely) ruled out such an explanation.
I have a lot of trouble imagining any experiment which could completely rule out every other influencing factor. There are silly thought experiments such as what if the world has been objective for the last x00 years - prior to that and after, the laws we've observed during the period would bend; things will become inconsistent.
It's silly and I don't even believe it, but it does show that science is incapable of completely ruling out everything.
The truth could be endemic mystery. Endemic insanely and terrifyingly simple endless levels of understanding. In a way science is nothing unless that's the reality.