Not having a physical address is a huge problem.
However the postal offices are now being closed.
I expect the same problem would occur with The US Postal Service shutting down branches?
Specifically, what problem do post office banks have that private banks do not?
Private banks close branches all the time.
New Zealand Post wants to shut its last 79 shops, but will not say when the last branch will close its doors
New Zealand Post said the move from stand-alone stores was old news and 801 of its 880 stores were already franchised.
Kiwibank originally had branches at the Post Offices.
Kiwibank is left with much less branches than there were Post Offices - branches often replaced by ATMs in the wall: it has "325 Kiwibank Limited Branch and ATM Locations" https://kiwibank.banklocationmaps.co.nz/en/nzl
I presume it will take much longer in the USA for the Post Offices to close. There is no reason something similar won't happen there - the same economic forces occur in the USA.
Note that most franchises are an in-store counter or small area set aside for the Post Office supplies. I can think of franchises in book-stores, stationery shops, pharmacies, and even a public library. I know of one Post Office branch of that 79 that remains open, however it is mostly run by unpaid volunteers including a friend of mine (no banking counter).