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.
Gotcha. I would argue back that the "under $1" isn't even a legitimate path of arguing, because the post office doesn't deliver mail for under a dollar either; you'd have to factor in the cost of the taxes that subsidize that under a dollar price.
Taxes don't really subsidize the USPS, losses are paid for by tax money but they're supposed to run at a profit, which is why they have to cut mail delivery or increase stamp price.
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.
Actually, my claim is not that the cost would be more than a dollar, but that private carriers would flat-out refuse to deliver to many locations in the US (and with good reason; it wouldn't be profitable). We already see this problem with the lack of internet connectivity in many rural areas.
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.
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.