Tom,
Several times posts haven't posted, as I rush off and come back to find them still in the box. The previous was one, so it was written before your recent post.
There are quite a few ideas tied up in your post, but the one I would address is what do we do, if our classical views are flawed? Epicycles being the classic example. It proved eminently possible to construct a mathematical model with the earth as the center of the universe, for the very simple and basic reason that from our physical point of view, we are the center of our view of the universe. Even the star maps of today would be meaningless from another location in the galaxy. This very success proved to be a profound stumbling block to further understanding. As is said of GR and QM, 'The math works, therefore it must be right.'
How do we go back and make sure all possible foundational assumptions are correct, when there is so much social and professional pressure to keep moving forward, even far beyond the point of rank speculation? I think even you have to admit that not only are there plenty of loose ends remaining, but many of those supposedly tied up bring to mind the old computing principle of "Junk in, junk out."
As for determinism, vs. probabilistic, it would seem we would have both, when the 'machine' is deterministic, but the input cannot be fully defined prior to an event. Otherwise, if we were to assume the laws of the machine are only just suggestions, all bets are off. Or we assume there is some God-like point of view, by which all input is known, even if the range of sources is effectively infinite. So it would seem the past is inherently deterministic, since all input has been factored into the outcome, while the future is inherently probabilistic, since there is no perfect set of potential input into events which have not occurred. It is only when one subscribes to some blocktime, or multiworlds view that the question becomes unanswerable.