I’ve noticed that al my large Goat Barns get regular ‘Milk spoilage’ flags and no one appears to collect or take any milk produced to anywhere so it all goes to waste. I built a Root Cellar next to them, and also a Cheesemaker and the man from the Cheesemaker will wall right past the Goat Barns and go a mile or few up the road to the Cattle Barns to collect milk and bring it back. I stocked the Root Cellar only with some test milk, and also grain and root vegetables to see if it either increased or decrease in milk stock and nothing happens.
It appears the granary I built does get used to get grain feed for chickens but I don’t think the Goat Barns are picking that up either, or the root veg so the whole idea of a local supply isn’t working with this work category. They’d rather walk miles instead.
I am also having the same issue. I have a root cellar right next to the barns with a minimum set for milk, but no one ever stores milk in there. I even get storage maxed on the barns but there’s room in the root cellar next to it. The cheesemakers will grab it but no one else will transport it out.