Hamilton wasn’t president. This law in fact may have been introduced specifically in order to prevent him from becoming so.
Just the opposite, the Constitution was written to allow foreign born men who were citizens of the United States when the Constitution was adopted to become President.
Keep in mind what the Presidential citizenship clause was all about. The English Civil War was the template through which the founders viewed many of their actions. They were especially worried about three bad things that happened in the English Civil War, when Charles I was overthrown by Parliament:
1. The successful rebel general, Cromwell, became dictator. George Washington solved that worry by returning his commission to the Continental Congress, and disbanding the Army.
2. The common people largely over through established society. That happened in some states, and the Constitution was crafted to keep the Senate (elected by State Legislatures) and the President (elected by electors, not popular vote) insulated from popular clamor.
3. The Monarchy returned, and executed or imprisoned many of the Parliamentary leaders. That was the big worry. The men writing the Constitution were afraid that George III would send one of his younger sons to America. That Prince would form a political movement, get himself elected President, and return the country to the British Crown.
In retrospect, that seems a foolish fear. George III didn't have any son with the leadership potential, and popular touch, to remotely pull off such a feat, even if Americans were open to such a thing. But to the founders, mindful of how Charles II had had Parliamentary leaders, who signed his father's death warrant, hung, drawn and quartered as traitors, it was a big worry.
Seen in that light, the restriction of the Presidency to "natural born" citizens was not meant to exclude a child born to American parents, in an American possession. It would be like excluding Albert Gore Jr., for being born in the District of Columbia, which is not a State. (And yes, I knew he was just raised there, but you get the point.)