The points about positivism are ignored largely because people can't resist going beyond what they can't really talk about. It's the product of a childish human nature. Probabilities and classical particles are simply an interpretation of quantum mechanics. The wave function is a mathematical fiction that encodes our uncertainty. It cannot be taken too literally.
In terms of your points about randomness, I'm not sure how one can even suggest causality or pretend one even produces a coherent model without assuming randomness is simply about ignorance. The only alternative is mysticism. I don't think people appreciate that chaos IS random and it doesn't just look random. The problem is that people have abused a term that is simply associated with various mathematical distributions. The reason why the term "essential randomness" fails to have any meaning is that one runs into an undefined wall when tracing the argument backwards. In other words, no explanation is not an explanation. To the extent you can't explain something, you don't have a theory at all. It's impossible to prove that anything is "essentially random."
Your point about uncountable numbers were addressed by Stanford mathematician Solomon Feferman and he showed that Cantorian mathematics had no application in the physical sciences. However, the proofs are not agreed to be sound but are debated. A mathematician can't even provide one example of a real number, so they are even fanciful from a mathematical standpoint. Additionally, even countable sets are debated by ultrafinitists, since they are associated with the completed infinite set of natural numbers. The main paradoxes introduced by Cantor, Godel, Turing, etc.. were the result of completed infinity (self-contradictory) and "diagonalizing" (Poincaire's impredicativity and paradoxical self-reference). It's really quite trivial and has no bearing on real mathematical truth or decidability properly defined. Incompleteness is a very poorly understood result and amounts to a parlor trick of no real consequence. It's an extremely easy result but the simpler proofs have not been seen by most people. It says nothing about the undecidibility of other problems (even major mathematicians fail to understand this).
Mathematics is seen to be separate from science but it cannot also be free from the law of non-contradiction. Additionally, I would argue that when mathematics leaves physical intuition too far behind, it isn't very productive. Mathematics is certainly only a map but it has to be a map of something. For instance, ZFC is said to establish contemporary mathematics but that explanation is completely backward. Math is largely discovered and later described in terms of axioms. Then we have the fatal blow to this entire "disease," as Poincare described it: axioms are sensitive in the same manner as chaotic initial conditions. There are concrete problems where different axioms produce different answers, such as coloring problems. We have to know all their applications ahead of time and so they only work like fitting a curve to the past. It's like trying to pull yourself up by your own bootstraps. ZFC axioms are certainly not obvious but in fact quite arbitrary and one may even find a different starting point for the same system. I don't buy the idea that math is set theory.
What I'm saying is that these disagreements point to poor logic and not equally valid ways at looking at the world in different contexts. I would not be so kind to science or mathematics. Both need to study logic.