The USPS has the massive benefit of government-protected monopoly on first-class mail. Even if it's not a direct infusion of funds, it's a major subsidy.
I don't know how you can prove that, postal services worldwide have been a government run business until very recently, and are still kept as a monopoly even after being privatized. Fedex cannot ship your normal mail, they are not even allowed to do it.
As for your claim that private companies will not be going in unprofitable rural areas, well even the USPS does not go there anymore (or much less often) : http://m.wfpl.org/?utm_referrer=https%3A%2F%2Fwww.google.com...
Not even close. If you live out of town, you don't get a mailbox. You get a PO Box in the post office in the closest town.