Yes the media makes them seem like part of the Gov, but they are not.
Most of the Fed profits are handed over to the US Treasury, so it knows who the boss is, but Congress is treated with contempt by the Fed.
" give me control of a nation's money and i care not who makes its laws" mayer amschel bauer
who gives a shit about a few basis points of profits when trillions go through the fomc and the discount window?