The only time a letter carrier is not required to deliver your mail is if there is a loose dog in the yard (yours or neighbors). Safety is the first priority. If you want your mail delivery uninterrupted bad weather, keep those sidewalks shoveled and sanded. The carrier shouldn't have to risk life and limb.
The Postal Service is a government agency and the rules and regulations are made at the Headquarters level in Wash., DC. No local post office (anywhere) has the authority "to run things pretty much their own way". That's a Federal offense and should be reported to the U. S. Postal Inspectors. They are the law enforcement arm of the U. S. Postal Service. They're the "good guys". My carrier in Colorado never skips a beat.