Different resources should be slightly easier to gather in different seasons. possibly a 1.2 bonus or similar.
This would give high level players a chance to micromanage gathering to optimise collection speed, while allowing entry level and hands off players to play as they currently do.
This will also make the "don't gather X" tools more useful moment to moment.
Clearance weather could provide a bonus to gathering goods that want to be dry stored e.g grain, herbs and roots
Standard weather could provide provide a bonus to gathering thematically water dependant goods e.g moss broccoli, berries and reeds
Storm weather could provide a bonus to gathering animal products, where the animal producing the goods are affected by the storm e.g insects and eggs
If it was made into a cornerstone, I think one simple design would be "Gathering buildings have X bonus during clearing, and Production (industry) have a X bonus during drizzle." I.e. stay inside while it drizzles, go gather stuff while it's sunny. Think that's flavorful and might be broad enough to justify the micro involved. A per-resource modifier on a cornerstone might be too complex and quite a mouthful, too.
I personally do not like that suggestion, as I think it introduces too much micro. If one wants to play optimally, one will be forced to shuffle the workforce with every seasonal change. This would introduce a lot of tedium in my opinion. However, I think the idea is conceptually sound - perhaps some variant of that idea could be made into a cornerstone, turning it into an opt-in decision whether one wants more micro management or not, and as a side effect of it being a cornerstone, the effect could be stronger than the here proposed 20%.