The Constitution permits the federal government to establish post offices. So, it is one of the few things that are legitimate for the federal government.
Nearly all of my mailings are ordinary mailings. For 45 cents, the post office will take my letter and deliver it across the country. Will Fed Ex or UPS deliver my letter for 45 cents? I have real doubts about that, although I’ve never tried. I’ve had no reason to look for a better price.
I’ve also seen the videos of private carriers treating packages like garbage. I recall the video of the fellow throwing a package (I think it contained a monitor) over a fence onto the concrete. So, private doesn’t necessarily mean quality or greater efficiency.
For me, the 45 cent stamp is a pretty good deal. I’m not sure I want to eliminate that option.
FedEx definitely does not compete on price: I believe that the very cheapest standard offering that they have for an ordinary mailing is about $10, and they do charge more for going across the country than across the street.
FedEx has a rate calculator at https://www.fedex.com/ratefinder/home, but I don't believe that it includes any fuel surcharges that may apply.
Instead, FedEx competes on performance. I have had much better luck with mail that I care about with FedEx than I have had with the USPS. I would be glad to pay the USPS $1 per first-class stamp if I could be sure that my mail would actually get there in a timely fashion. It would be a lot cheaper than what I am paying FedEx for the same service.