If there is an input "out of our reality", then our reality would not be predetermined nor would it be random. (Which makes sense for the simulation hypothesis, which I enjoy having that conversation with my colleagues)

When a system is closed, things become harder.

Chaitin's number represents (very roughly) the structure of any complete mathematical system and it can be shown to be random in the sense of Kolmogorov.

Still, I don't understand any of this even vaguely past the platitudes.

The crux is that randomness is indistinguishable from a large collection of pre-determined hidden variables (imagine rolling all the dice for all of the outcomes of Physics ahead of time), but not all of the behaviors of quantum mechanics can be accounted for by (local) hidden variables (Bell's theorem.)