This is more than quibbling - there are reasons why foreign-born citizens can’t run for President. I’m sure Schwartzenegger would have something to say about the whole ‘quibbling’ issue.
Throughout European history, it was common to have the ruler of one country be the head of state or outright ruler of another. Charles V was the easiest example of that, but all the intrigue of the Hapsburg Hegemony and the installation of rulers by the Pope, or one of the Popes as the case might be, this is what drove the idea of the US being led by Americans, and not Europeans (especially Kings or Queens).
The entire idea of nationalism was still a relatively new notion as late as 1540. You may be the Spanish King, and have an army largely of Germans, facing the French King, with his army of Spaniards and Swiss. The land in dispute may have been, geographically speaking, Italian, with the land populated by Italian speaking people. This was a common occurence in the 1500’s. The US was formed not 250 years later, philosophically.
The codification of the United States, electing a leader of its own citizens, by its own citizens, was novel for the day. Give it 250 years hence of unbelieveable prosperity, and poor study habits, and people’s heads get as soft as cheese about the subject of the importance of nativity in national leadership.