I don't know if I agree with your first paragraph. The weather model analogy is at least misleading since the corresponding thing in QM would be a mixture of states, not a superposition.
Not really. For the weather model, instead of recording my state as (20,80), I could just store it in memory as (-sqrt(20),sqrt(80))/normalization, with the prescription that to regain probabilities, I must square each entry. The fact that there are negative or complex numbers in the state of a single particle is not concerning. Only when you have entanglement for two or more particles is when something non-classical happens. Please see my other comments under my parents comment.