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.
If it's not a letter the government does not assert sole authority.
The privacy of a mail receptacle is yet another issue ~ has nothing to do with what is usually termed "the monopoly on letters".
Let's say you have curbside delivery. You want to receive other stuff from other sources so all you have to do is put out another box.