My belief is that stopping deliveries is a business decision, and all business decisions are made by management/owners. I would NEVER put that decision in the hands of a general employee. Your mileage may vary, and it should match your business philosophy.
That said, we are extremely cautious about weather and safety of our staff. We are quick to tell customers to expect slow-downs in rain or othr inclement weather, and to stop altogether when conditions look icy or dangerous. Same with customer properties that are treacherous. We protect our staff because they are the core of our success. We will make some tough decisions sometimes about weather that is simply unpleasant, and take some grief. But real dangerous and risk is serious business.
We always and often ask the drives about road conditions, and ask some of our outlying customers how their roads are. We are fortunate the customers are pretty understanding if we say it is too icy to drive out (afterall, they aren’t willing to do it themslves, eh?)