I still say they should have known that Hamilton was ineligible, due to his foreign birth.
You're incorrect.
All of the presidents up to Martin van Buren were foreign born.
They were born as British subjects, not American citizens...
Hamilton WOULD have been eligible to have been President of the United States had he run and been elected.
The Natural-born Citizen clause of the Constitution reads:
"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."
Note the second and third phrases. "or a Citizen of the United States, at the time of the Adoption of this Constitution..."
That's why the first 7 men to hold the office of President were eligible to be president, even though they were born British subjects.