If you ask me personally, I think the federal reserve as a non-executive non-congressional agency is insane, yes. But as a legal question I would hope the president has to establish the control he's exerting is actually in the executive branch before he tries to exercise constitutional powers of the executive branch that supersede contradictive law.

It would be pretty nuts if the president could just fire whomever he wants no matter what branch they are in based on a simple declaration he considers them to be in the executive branch.