Let them fail. The reality is mail volumes have declined because only junk mailers are sending paper to you, with the random form or card that you need a physical copy of or when the issuing agency won't do it electronically.
There are still places that do this, the credit union I've been with for the better part of 3 decades still hasn't electronically implemented 1099s.
The first thing I'd do if the postal system went private? Rip that damn mailbox off my house and throw it away. You want to send me junk? You pay.
Do I want the post office giving me an email address? Yah, that would have been fine in 1996, but now? I guarantee you that 99% of those would be abandoned because of spam. The only ones using them would be morons who type facebook in google after searching for it in bing. Besides, the post office would have gladly harvested that data in email accounts, that's why they wanted it. I don't care what suburban mom says, that's always been the case. The post office knows where you are and where the letters that go to you are, and have no problem whoring that out to any three-letter that asks.
The person fails to mention that, while there was intense lobbying pressure for banking, the FDIC largely made the trust issue obsolete. The higher interest rates paid by private institutions meant a falling deposit load. By most accounts, there was only 12% of the deposits, somewhere around $400M, in postal accounts by the end of the program, with some 10% of those probably being dead accounts with the average deposit being around $70. This was in comparison to just 17 years earlier at the end of the war, where deposit load was almost $4B.