To add to your point that the president is all-powerful: The power of the president has expanded by a general tendency for Congress to not only pass mandates but to do so with broad goals and large discretionary leeway. The president executes these mandates, and over 240 some years, there is just a shit ton of legislation that has expanded the administrative state. And the Madisonian system of checks and balances probably isn't anywhere near as strong to check the power of the president as the Founders may have wanted.
Eric Posner, legal scholar and a sort of political scientist, makes an interesting and compelling argument that that system of checks and balances isn't the chief constraint on presidential power. From the abstract (does anyone have access through the paywall?):
The presidents who routinely are judged the greatest leaders are also the most heavily criticized by legal scholars. The reason is that the greatest presidents succeeded by overcoming the barriers erected by Madison's system of separation of powers, but the legal mind sees such actions as breaches of constitutional norms that presidents are supposed to uphold. With the erosion of Madisonian checks and balances, what stops presidents from abusing their powers? The answer lies in the complex nature of presidential leadership. The president is simultaneously leader of the country, a party, and the executive branch. The conflicts between these leadership roles put heavy constraints on his power.
I recommend this episode of Freakonomics, a podcast, to hear more from the man himself.