The big sell of the postal service is that it costs the same amount to send a letter from San Francisco to Los Angeles as it does to send a letter from the bayou in Louisiana to Honolulu, Hawaii. In this, physical mail is the big equalizer.
The USPS is also required to deliver mail to even the most remote of places because if the mail system did not work for everyone then several things would start to break: collecting taxes or votes or anything that a normal citizen should require can be done through mail, at minimum.
Really, the only way to replace mail is to provide the same level of reliability and free-to-receive service for another communication medium such as telephone or internet.