If you want universal postal service to rural areas (not profitable markets), then simply provide it and understand it has a cost like the military and education (as noted above)?
Having said that, having a universal postal service requires physical presence in lots of remote areas. And those remote areas not only lack postal service, but also banking, tax services, and even why not grocery shopping. I do not see why we could not let the postal service provide those extra services, especially in the remote areas. People in big cities would likely have a different bank, they have access to a DMV near by, or supermarket. But, in a remote area, how a retired person can cash his social security ? renew his driver license ? Maybe there is a "service" opportunity that would make those remote post office more sustainable. I do not see why the postal office could not provide other services especially in remote areas where they lack. The physical location of the post office becomes a more valuable asset.
The structure also hints at problems with government run businesses- the PX system avoids all sorts of sales and property taxes and gets reduced or free rent. The commissaries do that and get direct subsidies as well.
Before any service members (or their spouses) go after me, no, I am NOT arguing for any change to the system, just pointing out how existing examples look and work.
The legacy system is based on the fact that it is public service, equally offered to all citizens, regardless of their gender, race, religion and location. So, it is not a business. A business would focus on the most profitable aspect of the postal service. It would likely not serve some areas, or even discriminate against some people.
I believe, the reason why USPS still has a monopoly in first class parcel mail, it is to satisfy people we claim that USPS should make money. Now, if it was clear for everyone that USPS should not be profitable (like military or education are not profitable), then it would make sense to break all USPS monopolies. Actually, I would argue the other way around, then let's USPS compete also with banks (account, loans, credit cards, stocks/bonds, CDs, etc.), title companies, check cashing business, etc. Which I believe USPS cannot currently. USPS could leverage its geographical coverage to bring banking services to remote areas. I would suggest to combine some services like DMV, so we could renew ID/driver/passports at postal office. We could pay taxes and fine. We could access notary services (titles), etc.