The case of a quantised version of a classical chaotic system has been discussed in many papers. If the system evolves in complete isolation then its expectation values don't closely match those of classical chaotic systems because of interference. But real systems on the scale where we observe chaotic behaviour don't evolve in isolation, they interact with the environment. That interaction introduces additional terms to the system's equation of motion that make the quantum and classical expectation values match one another on the relevant scales of length and time "Decoherence, chaos, quantum-classical correspondence, and the algorithmic arrow of time" by Zurek:
and many similar papers. It should be noted that this is not the same as taking the limit $\hbar\to 0$. Nor is it the same as simply letting wavepackets diverge, which would give results that do not match classical choatic behaviour at all. For example, in classical chaotic systems, the relevant behaviour is that trajectories that are very close together can exhibit very different behaviour over time. But in wavepacket spreading the whole idea of trajectories is a bad approximation because of interference terms. The trajectory approximation only works because of decoherence.