How do you explain FedEx and UPS?
I can see both sides of this argument.
I think GP's claim is not that private parcel services cannot exist, but rather that if they were to deliver first-class mail, they would charge more than $1 per letter.
They can afford ridiculously low postage because they have a government granted monopoly on mail so they can price their rates at a level to subsidize Remote, Oregon with profit from Portland. If they didn't have to deliver to the middle of nowhere they could probably lower prices significantly.
Which is why they need the monopoly, if I could set up a mail company in just the top 20 cities in the country I could undercut the USPS and then they'd suffer enormous revenue shortages because they'd still be mandated to deliver everywhere.
So if you believe that this sort of communication is a fundamental right for all citizens, than the service needs to be provided by the public sector. It's impossible to build a profitable business in the private sector out of it.