I don't like the precedent for that.
I don’t know what “precedent” you’re talking about. The only close thing I can think of, is when Sam Houston was considered for U.S. President during the 1850s, at least by some of his fellow Texans. Houston was born in Virginia and was, thus, a “natural born citizen,” but in migrating to Texas, accepting an commission in the army of that place, and certainly in accepting the office of president of that place, he had become a citizen of Texas. We, back in the United States, might have thought of him as a dual citizen. But, it could be said that, upon Texas’ annexation, he and everybody else there, got U.S. citizenship either brand new or restored. This is the precedent I had in mind.
I don’t think anybody born in Louisiana prior to its purchase, or born in the Mexican states of New Mexico or California prior to their purchase, or born in the Kingdom of Hawaii prior to its annexation, wgi was thereby made a U.S. citizen, was ever considered for U.S. President. But, according to the Congressional Act or the Treaty that was involved, the people of those places were not “naturalized” in the sense of the relevent section of the U.S. COnstitution regarding the qualification for the office of President. They became U.S. citizens by reason of their birthright in the places annexed. So, the only precedent we have (Houston) isn’t as close as we would like.
BTW the Constitution says that any person who was a Citizen of the United States at the time of the adoption of the Consitution, atherwise qualified, would be eligible for U.S. President. This meant, e.g., that Alexander Hamilton, who was born in the British West Indies, and “naturalized” a United States citizen (informal as that process was back then), was qualified.
With regard to persons born of U.S. parents overseas, or of persons born to immigrants to the U.S. while they are on the high seas, or of persons born in U.S. Territories such as the Canal Zone when it was a U.S. Territory, and of course any person born in the United States proper, they are American citizens by birth.