Let’s look at the Consitution.
Article II, Section 1:
The executive Power shall be vested in a President of the United States of America.
That’s literally it. No qualifiers or exceptions or elaboration. That’s a pretty strong argument in favor of a unitary executive.
But then, look at Section 2:
The President shall be Commander in Chief of the Army and Navy of the United States, and of the Militia of the several States, when called into the actual Service of the United States; he may require the Opinion, in writing, of the principal Officer in each of the executive Departments, upon any Subject relating to the Duties of their respective Offices…and he shall nominate, and by and with the Advice and Consent of the Senate, shall appoint Ambassadors, other public Ministers and Consuls, Judges of the supreme Court, and all other Officers of the United States, whose Appointments are not herein otherwise provided for, and which shall be established by Law: but the Congress may by Law vest the Appointment of such inferior Officers, as they think proper, in the President alone, in the Courts of Law, or in the Heads of Departments.
So at least the President’s appointment powers seem to be potentially circumscribed by legislation, and the existence of the Executive branch offices themselves are established by legislation. By extension, Congress also probably has the authority to establish who can fill those offices and what Executive powers can be delegated to those offices, but those Executive powers are still the President’s powers.
Personally, I’m with @Max_S on this, more or less. I don’t think the President can exercise powers delegated by Congress to regulatory bodies, even if those bodies are nominally in the Executive Branch and their members are chosen by the President. Those are Congressional powers, and the Congress has full and absolute power to delegate them to whomever they wish, and whomever they wish ain’t the President.
On the other hand, other Executive Branch agencies are exercising Executive powers on behalf of the President, and, see Section 1, all Executive power is vested in him. I don’t think Congress can require the President to delegate any Executive powers.
So (in my opinion), the President can’t act as his own Chairman of the Federal Trade Commission. The work of the FTC has to be carried out by actual Commissioners. But the President can act as his own Secretary of Defense. But then he actually has to physically do the job. He can’t give NotSecDef Jane Jones a “The Prez Sez” rubber stamp, and have her do the work. The President would have to actually be the one issuing the orders and signing the paperwork.