The list of "appointees" are provided by whom? And the fact that the entity itself is owned by private "citizens" pretty much makes it a private entity even with the false sense of oversight.
Quasi-government/quasi-private is how it is described by Wiki and several of my professors. Although it is far more private than any in government will ever admit.
The "ownership" is quasi itself. Banks are required to buy shares and they cannot trade them. The great bulk of the feds revenue after expenses is paid back to the treasury. It's not that private.