I have to agree with kfmfe04 on this. It doesn't matter how much I might like a particular ball, if I don't know the order of the balls inside the bag to begin with, even though I know which balls are in the bag and that they're in a particular order to begin with, then my ball pulling will produce a random distribution.
I suppose that the "to begin with" wrt quantum physics is it at the level of physicists' "pulling balls from the bag" -- since, as you seem to indicate, "no one knows for sure" what's in the "bag" that physicists are probing to begin with?
So, might one say that, unlike the circumstantial ignorance which produces the random distribution of balls pulled from a bag, quantum randomness is based on a profound, principled (via qm), and quite possibly permanent ignorance of the deep reality of nature.