> the postal service is a fully public agency, the fed is partially governed by private banks.

The Fed is fully governed by a federal agency. (The Fed Board of Governors).

The individual regional Fed banks are private-public hybrids, but they don't govern the system.

Nonetheless, given that the individual regional banks are have directors that are elected by the member banks, control over certain levels of fed policy are in a sense private, whereas that is nowhere true for the postal service. At least not to my knowledge.