Many decades ago as a despatch rider I asked a motorcycle shop for some waterproof gloves. They sold me some Oxford Aquaproofs. They were waterproof, completely! The water ran down my sleeve and into the glove , there to stay, until the beast was removed and upended to empty it. One of my fellow riders advised me to get some thin racing gloves that I could push inside the cuff of my jacket, which turned out to be a lot warmer and no less dry. Waterproof overgloves and silk undergloves for really cold days and the problem was solved.
Edit: forgot to make clear, I was a motorcycle despatch rider, not a bike courier! Sorry for any ambiguity.