Isn't the Federal Reserve supposed to be an independent central bank? So, independent of the Legislative and Executive branches? How is it independent if the President & Senate choose who leads the bank?
The appointments clause of the US constitution gives the president the power to appoint "Officers of the United States". Unlike the Bank of England, which was a private organisation that was later nationalised, the Federal Reserve was created by an Act of Congress, and the Board of Governors is appointed by the President, under the constitutional power mentioned.
In this sense, it is not independent of government, however, its decisions are not subject to Presidential review or approval, and appointments to the board are for 14-year terms and so span multiple presidencies.
Unlike many central banks, the board members openly identify with political parties, and appointments to the board are fought over party lines, as all appointments require the "consent" of the Senate.