You can not just waltz into JFK for a flight to London without a passport. You get you passport locally via filling out the proper papers. My son was born in Reading England county Berkshire. I took his English Birth Certificate and my passport and wife's passport down to the American Embassy in Grosvenor Square in London. A couple of weeks later I had his passport in the mail. I also obtained a British Passport for him which was quite legal via the same method. I submitted his British Birth Certificate with the mail in form from the local Post Office. A few weeks later he had his British Passport. Today he could not run for the Presidency of the United States as he is still a legal British Citizen. If he renounced his British Citizenship he could run for President tomorrow.
In 1972 I had a American military passport, it was stamped “and infant son” after his CRBA was filed. I flew home three years later, with that passport and him.
I doubt your son could run for President.