I grew up on a farm. Nothing is more disgusting than warm milk straight from the teat. It doesn't get any fresher than that.
I would take it from the cooler. It was always pumped from the cow straight to the cooler. And my grandpa and uncles docked all the cow's tails, so they wouldn't get swatted in the face. Then PETA found out farmers across the state were doing that and raised holy hell so they can't do that anymore