> some other federal agency that the President has 100% control over
Neither Civil Servants nor Senior Executives can be directly terminated by the President without due process. Only appointees can be fired on a whim, and their replacements must be confirmed by the Senate. The only exception to this is "recess appointments" made when the Senate is not in session, but those appointments are temporary, and Congress can stay in session as long as needed (until the next seating after an election) to block these as long as both Houses agree to do so.
Many federal agencies today are actually governed by boards of commissioners, with limits on the number of them that can be from the same party.