No, the essential service is mail delivery under a dollar to anywhere in the United States. That is a fundamentally different service than what is provided by Fedex or UPS.
Simply because it's different doesn't mean it's essential. Is there an inalienable right to mail delivery whose rate structure ignores distance? Why doesn't the same right protect people from paying a premium for mailing heavy objects?
The essential service is universal mail delivery for a uniform (not cheaper to mail something to Chicago than Bumblefuck, WY), reasonable cost. Could/would private companies do that? Reply hazy, try again