As I see it, the question is whether it is in the best interest of the country that all citizens have regular mail service at an affordable price. Maybe it isn't now. But, it used to be considered important. I can remember when anybody could send a letter any place in the US for 3 cents. Of course it's more now, but so is everything else and the principle is the same.
Could a private company do that and make a profit? It would be tough. Some addresses are pretty remote and delivery cost is almost the same whether it is one letter or a bundle of mail. Several companies competing in the same area probably couldn't. So, it would probably have to be a monopoly, and we know how pricing and service works out in that situation.
Mail carriers make a decent salary, but it's not exorbitant. A private company would probably be closer to minimum wage. Turnover would probably be higher because it's not an easy job. Like the saying goes " Neither snow nor rain nor heat nor gloom of night ."
Would congress let the postal service die? I sincerely doubt it. They may rant and rave and call for an investigation, but there are too many constituents and businesses who would not like to see it go. For one, it would be the death knell for the magazines that still remain. There is a reason that most small businesses ship packages by USPS rather than one of the other carriers. It's cheaper. Even UPS uses the postal service for final delivery of some of their packages.
Rates will no doubt go up. I wouldn't be surprised to see one dollar stamps. That seems extreme, but there was a time when I would never imagine $3+ a gallon for gas either.
Don