No we fucking dont and fuck no "electing a defacto king" was never the fucking intention of the founding fathers. George Washington was famously insulted at the notion that a US President would ever be compared to a King.
You're generally correct, and I could have explained more thoroughly but it was already a big post, so let me autism more.
A more accurate definition of what the Founders sought would be, drum roll please,
Dictator. Federalist 70, written by Hamilton but you can bet your ass that Madison is involved considering this is the argument for his baby, explicitly states this; "Every man the least conversant in Roman story, knows how often that republic was obliged to take refuge in the absolute power of a single man, under the formidable title of Dictator, as well against the intrigues of ambitious individuals who aspired to the tyranny, and the seditions of whole classes of the community whose conduct threatened the existence of all government, as against the invasions of external enemies who menaced the conquest and destruction of Rome."
Now, what about the powers themselves, and what actually makes up the difference between a king and any other executive, as extended by English and historic understanding of the time?
Federalist 70 again states; "The ingredients which constitute energy in the Executive are, first, unity; secondly, duration; thirdly, an adequate provision for its support; fourthly, competent powers. The ingredients which constitute safety in the republican sense are, first, a due dependence on the people, secondly, a due responsibility."
In this we see that the safety required for a republican government is that the powers of the Presidency do not derive from tradition nor god, but instead from the people, and that the responsibility lies not towards the continuation of a crown but instead that of the people. The powers of kingship are there, the baseline ethics however are somewhat different. Realistically, every human society of sufficient size has to form a strong executive, thus it's the memetic evolutionary equivalent of crabs. George Washington, and others, set the
social status of the President as something not to be king, but the actual powers and limitations thereof? Even at the time, kings already had checks and balances against them and could not completely alone, whether that be the lords being lords, or divesting power explicitly to the commons to form law as early republics formed, IE English and later British Parliament, but kings at the time still retained quite a bit of executive function.
Dictator is a far more fun term however, so lets use that.
