Fedex can (and do) deliver letters. They cannot deliver first class mail or use the mailbox, the interior of which is technically property of the US post office.

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.