Hasn't this been worked around, though? I can take a letter or other paper documents to FedEx, and they will ship it. It will be more expensive than USPS, though.
I guess I don't understand the logistical difference, then, because that sounds like a semantic difference to me. What stops FedEx from saying "we ship this parcel for the same price as USPS ships a letter"?
One major restriction is a prohibition on offering delivery on a regular (daily) schedule. The relevant statutes also call out restrictions on price of postage as a multiple of the going postal rate, as well weight of the parcel. The USPS also has exclusive access to mailboxes.
USPS has to get paid the going rate for postage. But given that no private carriers are within a factor of 10 of USPS pricing, that seems like it's not the barrier here. (Also: much easier to make the argument that the $1.00 letter service from FedEx could be $0.45 if Congress would update the relevant law.)
This really looks like the private carriers have carved off the profitable part (rapid delivery) and leave USPS with the hard, unprofitable part (delivery to 100% of addresses, for $0.55).
I'm not sure how you reached your interpretation. The USPS is not barred from urgent delivery and has a competing product. They're just not as good at it as the private companies are.
As for regular mail, they don't offer that because it's impossible to compete when you have to pay your competitor for every sale you make.
It doesn't have exclusivity for letter service. UPS, FedEx, and DHL (among others) will carry letters across the country for you.
Yes, it's going to cost ~$10 to roughly match the service speed of a USPS first-class stamp. But it's available.