“I always thought that a big reason for the “natural born citizen” requirement was to keep Alexander Hamilton out of the Presidency. He was a fan of strong central gov’t to the point of possibly being a monarchist.”
This is correct.
Didn't matter where he was born. He was grandfathered in by being a US citizen when the constitution was adopted
No person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty-five Years, and been fourteen Years a Resident within the United States.