Others have already mentioned this, but OP seems to completely miss the fact that USPS is obligated to perform its affordable, uniform-rate deliveries to rural areas.

https://www.ips-dc.org/the-public-postal-service-and-rural-america/

Rural areas account for 88 percent of the geographic area served by the Postal Service, but just 16 percent of the U.S. population.

Some 57 percent of post offices are in rural areas and while they serve a vital purpose, nearly two-thirds of them have not been profitable in recent years.

Given that most of the population lives in cities, it makes sense that competing operations would form to provide services that can be cheaper, faster, and capable of shipping packages that are extra heavy, large dimensions, or require refrigeration. Specifically because these private shippers rely on USPS for last mile delivery or do not service rural areas at all, they are able to operate with much better margins. These companies grow into megacorps because, as OP mentioned, economies of scale. If someone needs to specialty ship something they are going to prefer to use a well-known business with a proven track record that services the widest area. Once a company is servicing an area, it becomes more efficient and therefore profitable to increase deliveries within that area, creating extra incentives for a shipper to grow.

I'm not here to say that USPS is flawless, but it is inherently at a disadvantage for urban delivery due to the losses it incurs to fulfill its obligation to service the country as a whole.

This is getting long-winded for a showerthoughts response, so I'm just going to link information regarding the additional obstacles USPS faces:

https://govtschemes.org/usps-operational-restructuring-plan-update/

https://govfacts.org/government/federal/independent-executive/usps/understanding-the-biggest-problems-facing-usps/

(I realized after reading that govfacts.org are AI generated articles, but there is good information, so I am still including it)

A fun little doc about your selected shipper(USPS), their corruption and pursuit of profit.

A postal worker is harassed and fired for standing up to a toxic culture. His son’s powerful documentary exposes how the USPS was systematically dismantled for profit — at the direct expense of the people who keep it running. A gripping true story of courage, corruption, and a system that turned on its own.

https://www.youtube.com/watch?v=NXsP1Gngd4w

whats really annoying is shipping would not even happen to rural areas for a decent price without the usps in the us. Its one of the bigger subsidies. Honestly same with electrification without the gov pushing it. Everyone dumps on the post office but from my experience they had the best service everything else being equal. Which its not given the republicans love to try and hamstring the operation. I don't agree private should not be allowed or anything but the governement should always be one with strong privacy protections. They should do the same online with the very same protections.

The thing is, theres not even a point in banning private competition. I believe somehow it got insinuated that my shower thought also suggested banning private competition, but that wasn't my intention.

The USPS is subsidized, tax exempt, a subsidiary of a bigger organization, and has built in economies of scale. If someone can compete on that super skewed playing field, more power to them! A side benefit is that private competitors being able to win on price or service despite USPS's huge advantages all but proves the USPS is dropping the ball, corrupt, or both; it's a built-in alarm.

How does this work on an international scale? If our hypothetical shipper flies to another country, do they waste time and fuel on an empty return flight?

I understand why health care should be socialized, but why should package delivery be?

Why stop there, maybe the government should have a monopoly on hair salons and pizzerias since everyone uses those too. /s