If you actually had some business experience you would know that bills can in fact be delivered by private carriers. As can legal papers and a host of other things. Yes the postal service is cheaper but the functionality is available with private carriers.
Anything you can deliver by first class mail can also be sent by private carrier. That is not a functional monopoly. Yes it will cost you more and the law is structured to ensure that it costs more but you can still do it. The reason it costs more is that private carriers are not subject to the universal service obligation of the postal service and if they were allowed to only serve densely populated areas which can be done at a lower cost the postal service could not provide subsidized mail service to rural areas. Now you can certainly argue that the postal service should not subsidize rural areas but that little chance of being enacted into law.
Now the reason that I used surplus with reference to the USPS instead of "profit" is because the USPS is a government agency not a business. Whats funny is that when you're trying to make one set of points you refer to the USPS as a government agency but then you try to talk about profit in a different place.
I'm going to point this out again since you don't seem to be willing to acknowledge it when you make your comparisons. The USPS is not the same as a private business. It has mandates and requirements imposed on it by Congress that no private business has to operate under. That is in fact the major source of it's current troubles. A private business can adapt quickly to changing circumstances. The USPS has to get Congress to agree to changes and when one party in Congress actively wants to destroy the USPS in order to target public sector unions it makes things difficult.