I think that's different. Chaos still uses classical probability and the randomness is just ignorance of underlying initial conditions. This is very different from QM.
I have. They still don't make Quantum Theory and Chaos similar. Rather for some systems QM can motivate ergodicity as well as classical chaos can. However that doesn't mean Quantum Probability and Classical Probability are alike simply because they give similar behaviour for certain systems for one specific limit. Their representation theory is completely different.
Chaotic systems don't have a Kochen-Specker or PBR theorem.