USPS does deliver packages. Before Amazon had their own trucks, they were mostly just USPS. Given the dominance of online shopping, how have they been consistently losing money? Were they subsidizing Amazon?
There is also UPS Surepost and Fedex Smartpost. AFAIK these are all charged at a lower rate for dropping pre-sorted packages off at the local PO for last-mile delivery. But of course such last-mile delivery is the expensive labor-intensive part.