anyway, it seems that the HUP is a QM result, and not a general one? So when can we know that the the system is "quantum enough", so as to apply the HUP ?
That's the difference between the quantum and classical systems. The ZPE means there is a lower limit to the uncertainty that doesn't exist in the classical system.
The uncertainty exists in classical and quantum systems. It's just a statistical property. But in a classical system the uncertainty can be made arbitrarily low. The difference is that in a quantum system the uncertainty cannot be made arbitrarily low.
@JackRod quantum particles don't really have a path. They don't have a well defined position due to the uncertainty principle, so they can't have a well define trajectory. They have a sort of average trajectory.