What OP is discussing was called a milk round. When they stopped being a thing depends on what country you are talking about. I can’t speak for the U.S. In the U.K. you can still find job advertisements for milk man positions or milk rounds. In Australian major cities they were common through to the end of the nineteen eighties. After 1990 they start to focus on deliveries to businesses and offices.
In Australia, the deliveries were originally by horse drawn carts, (or milk floats) but by the 1980s these were largely replaced by motorised vans. Although vans could be refrigerated and travel faster from the dairy to the start of the round they required more work by the milk man (or two workers) as a van only moves while a driver is in the driving seat but a horse can draw a cart from one stop to the next relatively unattended while the milkman is delivering to the front door.
In Australia the reasons they are no longer a thing are commercial. They stopped generating a profit. Reduced family sizes, wider motor car ownership, increased home refrigeration and the wide spread of supermarkets reduced the demand for daily deliveries by enabling families to purchase milk easier with their weekly shopping and store it until used. Note that daily bread deliveries also used to be made in a similar way by horse and cart but these stopped earlier.
The reference below is to a display by the National Museum of Australia which addresses parts of your enquiry although it is restricted to horse drawn milk rounds.