Laplace's Demon has been controversial for all its two-hundred year existence. ...No, the uncertainty principle has nothing to do with Laplace's Demon. The Schroedinger equation is deterministic, but wave solutions exhibit the uncertainty inequality anyway.
Physicists interpret the uncertainty principle in different ways, but it suggests that Laplace's postulate cannot literally be true. Perfect predictions are impossible if the universe itself is random.
Fortunately, weather does not require quantum mechanics for us to study it. It happens at a molecular (rather than an atomic) level, and molecules are much too large to be discernibly impacted by quantum physics. Moreover, we understand the chemistry and Newtonian physics that govern the weather fairly well, and we have for a long time. [p.113-4]
Molecules cannot be too large to be impacted by quantum physics. Most of the molecules only have two atoms, so the idea that quantum mechanics affects atoms but not molecules is silly.
Yes, meteorologists model air as an ideal gas of non-quantum particles, but Silver is leading up to an explanation of how chaos theory put limits on predictability, but those limits are ultimately quantum mechanical.