...does not exist. The Fed is a government entity whose policy-setting body (the Board of Governors) consists entirely of Presidential appointees confirmed by the Senate, and who can be removed by the President (at least “for cause” based on a collateral reference, and possibly without cause though that's murkier, though no explicit boundaries exist on the qualifying causes.).

It is “independent” within the government (has it's own policy-setting body which exercises directly-delegated powers, and whose members have terms not tied to Presidential terms and at least arguably cannot be removed by the President without cause), not independent from the government.