What did the founders really mean by "unitary executive"?

I'm cross-posting from material I wrote for the Wikipedia entry on Unitary executive theory, pointing out what a crock it is:

Critics of the theory point out that the Constitution grants Congress the exclusive power to "make all Laws which shall be necessary and proper for carrying into Execution... all... Powers vested by this Constitution in the Government of the United States, or in any Department or Officer thereof"; that the Constitution grants Congress the exclusive power "To make Rules for the Government and Regulation of the land and naval Forces"; that the Constitution specifically obligates the President to "take Care that the Laws be faithfully executed", where the "Laws" are defined as that which the Congress has the exclusive power to pass.[3] [4] They have also indicated that in every single Supreme Court case involving a statutory restriction of the power of the President, the statute has been upheld, including several in which the statute was only held to imply the limitation on Presidential power, let alone explicitly limit it; [5] that the phrase "unitary executive" that was discussed in the Constitutional Convention referred merely to having a single individual fill the office of President, as proposed in the Virginia Plan, rather than have several executives or an executive council, as proposed in the New Jersey Plan and as promoted by Elbridge Gerry, Edmund Randolph, and George Mason [6] [7]; and that the Constitutional Convention debates show that the Founders' primary concern behind whether to have a single executive or an executive council was to choose the one that would ensure that the executive would be relatively weaker and more easily restrained by the legislature; that those who argued for a unitary executive advanced the argument because they considered that the best way to limit the executive’s power and keep it subordinate to the legislature, in opposition to arguments that a plural executive would support the executive’s independence; and the term "unitary executive" was thereby bound up with the intention of keeping executive power checked and restrained. [8]

# [3] U.S. Constitution
# [4] Letter to Congress regarding FISA and NSA, Bradley, et. al., January 9, 2006; see cases listed therein.
# [5] Letter to Congress regarding FISA and NSA, Bradley, et. al., February 2, 2006; p. 5 (e.g. “The argument that conduct undertaken by the Commander in Chief that has some relevance to “engaging the enemy” is immune from congressional regulation finds no support in, and is directly contradicted by, both case law and historical precedent. Every time the Supreme Court has confronted a statute limiting the Commander-in-Chief’s authority, it has upheld the statute. No precedent holds that the President, when acting as Commander in Chief, is free to disregard an Act of Congress, much less a criminal statute enacted by Congress, that was designed specifically to restrain the President as such.” (emphasis in original) – 14 legal scholars including the current dean of Yale Law School and the former deans of Stanford and the University of Chicago law schools.)
# [6] Ralph Ketchum, ed. The Anti-Federalist Papers and the Constitutional Convention Debates (Signet Classic, 1986), p. 67 (“MR. [James] WILSON entered into a contrast of the principal points of the two plans [i.e. the Virginia Plan and the New Jersey Plan]… These were… A single Executive Magistrate is at the head of the one – a plurality is held out in the other.”)
# [7] Robert Rutland, ed. The Papers of George Mason (3 volumes, Chapel Hill, 1970), vol. 3, pp. 896-898; Ralph Ketchum, ed. The Anti-Federalist Papers and the Constitutional Convention Debates, pp. 47-49 (“If strong and extensive Powers are vested in the Executive, and that Executive consists only of one Person, the Government will of course degenerate, (for I will call it degeneracy) into a Monarchy – A Government so contrary to the Genius of the People, that they will reject even the Appearance of it. … If the Executive is vested in three Persons… Will not such a Model of Appointment be the most effectual means of preventing Cabals and Intrigues… Will it not be the most effectual Means of checking and counteracting the aspiring Views of dangerous and ambitious Men, and consequently the best Security for the Stability and Duration of our Government upon the invaluable Principles of Liberty? These Sir, are some of my Motives for preferring an Executive consisting of three Persons rather than of one.” George Mason, Constitutional Convention, June 4, 1787)
# [8] Ralph Ketchum, ed. The Anti-Federalist Papers and the Constitutional Convention Debates, pp. 42-43 (“MR. [John] RUTLEDGE… said he was for vesting the Executive power in a single person, though he was not for giving him the power of war and peace. A single man would feel the greatest responsibility and administer the public affairs best. MR. [Roger] SHERMAN said he considered the executive magistracy as nothing more than an institution for carrying the will of the Legislature into effect, that the person or persons ought to be appointed by and accountable to the Legislature only, which was the depository of the supreme will of the Society. As they were the best judges of the business which ought to be done by the Executive department, and consequently of the number necessary from time to time for doing it, he wished the number might not be fixed, but that the Legislature should be at liberty to appoint one or more as experience might dictate. MR. [James] WILSON… The only powers he conceived strictly Executive were those of executing the laws, and appointing officers, not appertaining to and appointed by the Legislature. MR. [Elbridge] GERRY favored the policy of annexing a Council to the Executive in order to give weight and inspire confidence. MR. [Edmund] RANDOLPH strenuously opposed a unity in the Executive magistracy. He regarded it as the fetus of monarchy. … MR. WILSON said that unity in the Executive instead of being the fetus of monarchy would be the best safeguard against tyranny.” Constitutional Convention, June 1, 1787; see also comments by George Mason, previous footnote.)