The letter mail monopoly covers letters as the general public imagines them, AND checks. It does not include non-communicative correspondence ~ handbills, broadsides, newspapers, magazines, and so on.
The Postmaster General is authorized to grant exceptions ~ which is why you have express letters handled by FedEx and UPS (and others).
At the same time for purposes of security the law allows the Postmaster General to require that intended recipients of mail maintain a receptacle not available to others. Technically speaking, you are the guy who keeps private delivery services out of your mailbox, not the post office.
No, sorry. There is still a government monopoly on mail delivery.
Remember, FedEx and UPS *CANNOT* open your mailbox to stick a package or overnight-delivery envelope inside. It is ILLEGAL. They cannot even stick it in a mail slot in your front door.
That looks an awful lot like a monopoly to me.