A number of presidents have had parents born outside the United States, James Buchanan for instance. I haven’t seen a reference that his father was naturalized before James was born.
Exactly.
That was addressed in writing of the time of the founders. The founders stated intent was to protect the rights of US citizens, people sent abroad on America’s business & travelers.
In the founders writings it is clear that they did not want anyone who was subject to foreign influence by prior citizenship or dual citizenship to be President or Vice President.