All the unincorporated places I'm familiar with in the mainland US are under the jurisdiction of federal, state and county governments, but not a city or town government. I'm sure there are examples of more sovereign towns... US Postal Service is federal and required to service the whole country. Lots of cities and towns manage utilities like power, sanitation, etc. and would probably exclude unincorporated properties from access. People just buy them from county governments, private companies, etc.

I think the primary idea of an unincorporated town is that it isn't incorporated as a municipality with its own government?

In places like Montgomery County, MD (which doesn't have incorporated individual towns) it would be very confusing if all the mail had to simply be addressed to "Montgomery County" despite the fact that almost everyone refers to its individual unincorporated areas as if they were actually separate towns.