Charlie, the USPS was running billion dollar deficits before the PAEA was enacted in 2006. Yes, it’s cheaper than Fedex and UPS, but that’s because debt is subsidizing its service. If there were a federal Pizza Agency losing billions a year while providing pizzas for 50 cents, it would be cheaper for consumers while inefficient overall.
Any economist will tell you that the classic case that requires public provision of a service is public goods. These are goods which cannot be consumed by one party to the exclusion of another (“non-rival”) and cannot be provided selectively to paying subscribers (“non-exclusive”), such as clean air and defence. Mail has neither characteristic, and the case for public provision of mail is no stronger than that for public provision of pizza or any other private good or service.