Traditionally, the state has certain duties it needs to perform for every member of the population.

Passports, driving licenses, polling cards, draft registration, pensions, company registrations, patents, copyrights, court summons, speeding fines, inheritance, tax paperwork, census, etc etc.

It’s much simpler to perform these duties if you have a means of communication that can reliably reach every citizen.

I'm not sure I'd put "reliable" in any description of the USPS. I get my neighbors mail in my box often. I can only assume some of my mail gets delivered to them as well.