The Post Office also says that rural delivery would stop. That's nonsense, of course, but people in sparsely populated areas might have to pay more for some services, just as city dwellers have to pay more for fresh vegetables and firewood. The free market would reduce the difference to transportation costs, however, thanks to arbitrage and entrepreneurship, and there would be constant competition to make transportation cheaper. The Post Office also claims that only the U.S. government can secure our privacy and guarantee access to the mails. But this is Newspeak.
Government is the great invader of our privacy, mail and otherwise.
In the 1970's, the CIA routinely opened mail. And the same thing is happening now to opponents of the administration's foreign policy. As to freedom of access to the mail service, the Post Office frequently claims the right to decide what can be mailed. But there is no such thing, only the natural tendency of people who want to live off the taxpayers through monopoly to claim there is. If any monopoly were actually natural, it wouldn't need a government gun to enforce it.