One issue with having the Constitution being written to permit foreign-born individuals to become president was that at the time it was written, there were a lot of conspiracy theories floating around and general paranoia among the population and the framers about European monarchies trying to subvert the government in an effort to destroy a republic coming out of a successful rebellion lest it inspire their subjects to overthrow them or because they wanted America for themselves (with an added revenge motive for the conspiracy theories and paranoia concerning Britain.)
IIRC, one of the most feared forms of such subversion at the time was a scenario involving a wealthy foreign aristocrat who was really a foreign agent emigrating to the US, becoming a citizen, and through a combination of demagoguery and buying his way into office, would effectively gain control of the country and promptly betray the US to whatever monarch they were supposedly working for. The requirement that a person must have resided in the US for 14 years was in part intended to reduce the possibility that a natural-born citizen with presidential ambitions could be corrupted to play such a role, while the language about someone who was a citizen at the time of the adoption of the Constitution was pretty much inserted for the purpose of allowing Alexander Hamilton (born in Jamaica) a shot at the presidency.
The conspiracy theories and paranoia that fueled those fears and inspired the exclusion of naturalized citiziens from the presidency only really started to die out after the War of 1812, but even after then, the challenge for finding a good time to so amend the Constitution would involve finding a point where not only there wasn't any great wave of anti-immigrant and/or nativist sentiment, but that the issue wasn't clearly linked to a specific individual in the government or who would otherwise be an attractive candidate for one party, for the issue to come up and the questionable utility of the restrictions since the paranoia about foreign subversion they were designed to prevent turned out to be unfounded.