Most of the reasons that it isn't profitable relate to problems that the USPS already has to solve as part of their core function. It doesn't make sense for a bank to invest in physical infrastructure in the middle of nowhere, the cost of the building would erase any potential profit. The post office already has a building there though (and staff, etc). A significant amount of the up front expense, as well as a decent chunk of the ongoing expenses, is already going to be spent anyway.
So for these five or six people in rural areas that can't get a bank account, and aren't complete deadbeats, and can't get to a bank as easily as they can get to a post office, the USPS has a cost advantage in serving them. Okay. I just don't see numbers that make this a big source of free money.
How can it act as both a public service and source of free money? Even if it made a small profit doing so, it would be less than ramping up their most profitable areas and would have an opportunity cost.
(If you mean they should do it purely as a public service, that's a fair point, but you're writing like it's an easy way to improve their financial position, which doesn't make sense.)