First class mail is a "monopoly" as someone downthread noted, but not the kind that anyone trying to make a profit would want. The post office cannot set prices on first class mail (that's done by Congress) and needs to deliver mail everywhere at the price Congress has mandated, even when it is done at a loss.
I just learned this, but yes, that appears to be the case! The private express statutes[1] appear to do just that.
In 1971 a private corporation, the Independent Postal System of America, offered to deliver Christmas cards for five cents each — three cents less than the U.S.P.S. rate. They were stopped by a court injunction.
In 1966 the CF&I Steel Corporation, frustrated by the quality of postal service between their Denver headquarters and their plant in Pueblo, hired an armored-car service to deliver the mail. After five months of operation the service was halted by the Denver Post Office, and "at the Post Office’s suggestion" CF&I paid two thousand dollars toward back postage..
Because letters are paper but so are magazines, newspapers, etc that I know are delivered by non-USPS mail carriers.
It isn't a very persuasive argument to destroy cheap post for the masses to... let investors make money. As it is if usps was incapable of fulfilling its duty private entities could enter the market at way higher prices and if useful would see business.