There are a small set of suppositions needed to show the power of the executive branch is in the cabinet.
The pro-Federalists wanted
to create a system of government that used the precept of consent to be governed
that maximized check and balances
that no person had uncheck power
that minimized the changes put upon the states
If they could make a system that met these requirements, they would do that.
If describing the system with appropriate 1800's era words would match the words in the constitution then probably the constitution and that system are the same
If X# actions make more sense with the revised definition compared to the current understanding, then its beyond a reasonable doubt that the revised understanding is correct.
Citizens of small absolute monarchies and dictatorships do not need to ask questions. These governments have a simple law, "Do what the boss says". For republics with a hierarchy of bosses, that changes to, "Do what the bosses wants". The test for an action's legality becomes, "When the bosses are happy the action was the legal. When the bosses are not happy it was criminal". The difficult part is knowing the future.
The authority to act as a lower-level boss comes down the tree structure hierarchy with the obligation and responsibility to keep the boss satisfied and content enough for the boss to pursue happiness. There is an ultimate authority and there are some penultimate authorities with the penultimate responsibilities. Written rules, regulations and laws improve the chances for guessing correctly. Accurate and easy to use laws of nature i.e. laws of physics and proclamations from the ultimate authority significantly reduce the number of wrong answers to consider. The odds of a correct guess about the future are greatly improved. But guesses do not make exceptionally great nations.
Governance is better when the ultimate authority to act comes from the bottom up. The top of the hierarchy has work to do because overall the obligations and responsibilities of the government do not change. But some adjustments are needed because the ultimate authority (every person) and the bosses with the penultimate responsibilities (the executive cabinet) have communication issues. Timely reports and timely orders cannot occur for large populations.
Effective communication also cannot occur for small populations that want to pursue happiness because listening to reports and giving orders is too time consuming. Then there are the children who are not old enough to know the meaning of giving consent to be governed or effectively participate in conversions about world affairs. Persons who are part of a civilized group (a people) ought to be represented even if they cannot give their consent to be governed. The people ought to have the civil liberties provided by their government.
There are multiple ways to mitigate the communication issues caused by having 300 million constituents being a single ultimate authority. (E pluribus Unum) The US Constitution is one of those ways. At the time it was written there was the only way that could be explained and ratified.
The primary reason to make the constitution complicated was to prevent the unitary executive (any individual) from having the ultimate authority to command those with the penultimate responsibilities (cabinet members). This can only be that the cabinet members as a group have all of the authority except to sign and announce the decisions. The president needs to nominate people he will generally agree with because they are going to tell him what orders to sign. The authority starts from the bottom and goes up to where the "buck stops" for sign-off.
Same system as above with emphasis on authority being handed off by election boards and Senate confirmations.
The state kept all of the control except for authorizing budget.
The President and Vice President chair the meetings sessions. The secretaries and Senators have all of the authority.
The President can veto spending but does not have much authority until provided the cabinet.
The processes unused in the Senate and the Cabinet would be initially similar and then only diverge as required. So, the President would not vote except in a tie during his cabinet meetings. He would not sign proclamations or bills without the advice and consent of the cabinet.
It seems backward because it has to be backwards to achieve the no individual with ultimate authority except during Senate declared times of war,
1862 letter to Horace Greely
Jay Treaty cabinet meeting burst of anger
Alien and Seditions Acts 1798
refuse to remove political opponents
sharply different after presidency
Louisiana Purchase
Embargo Act
national bank
war 1812
failed to lead his cabinet
Bonus bill 1817