Also we kinda want to keep territories like Alaska inhabited because geopolitics.
And we kinda want the people living in those places to be able to get their official government documents (driver licenses, court notices, etc).
Then there's this idea of voting by mail, which isn't something that should be a privilege, but that's too complex for our discussion.
To translate one of the reasons for USPS into words you may understand: you know how two-factor authentication requires that second factor? The USPS provides a means for the gov to get that second factor into the hands of the people.
A third party would be a man-in-the-middle.
So consiser USPS a part of the government's 2FA infrastructure and messaging system with SLA and proof of delivery.
That, unlike your company's authentication infrastructure, happens to pay for itself.