https://www.law.cornell.edu/cfr/text/39/320.6
There are minimums they can charge and some speed requirements.
https://about.usps.com/publications/pub542/pub542_ch5_001.ht...
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).
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.