That's the theory of the Unitary Executive, made famous by Bush's OLC head John Yoo.
But there are statutes that delegate powers directly to agency heads and cabinet members, and not to the President, others that expressly limit the President's authority over agencies, and there are independent agencies that the President can appoint people to but cannot directly control --- despite residing in the Executive Branch.
Also, the Constitution gives Congress the authority to vest the appointment of "public ministers, consuls, and all other officers of the United States" directly to the heads of departments, explicitly bypassing the President, who is mentioned earlier in that sentence of the text.
Ultimately I think you're mostly right in one sense, and the President has so much authority over the DOJ that resignation is the only real authority the AG has --- which explains things like the Saturday Night Massacre --- but that the day-to-day authority of the AG and the leverage they have over the President by threatening to resign makes the balance of power here tricky.
Again: John Ashcroft thwarted not just Bush but the real President at the time, Dick Cheney; so far as we know, he didn't threaten to resign in protest, but rather just refused to authorize STELLARWIND. And, unlike the Saturday Night Massacre, Bush's party controlled Congress, and was not hostile to what he was trying to do.