You`re so close, yet so far removed from original intent:
Had the original framers intended for any born Citizen to be eligible to the office of President, they would not have included the word natural in the clause. Additionally, had the framers of the 14th Amendment intended to declare that every person born in the country was a natural born Citizen, then the 14th Amendment would contain clear and manifest language to that effect. But it doesnt. Therefore, each clause must be given separate force and effect.
There are two classes of Citizens in the United States:
1.) Natural Born
2.) Naturalized
They were familiar with the phrase "natural born subject" or "natural born citizen" as equivalent to "native-born subject" or "native-born citizen" or "subject from birth" or "citizen from birth."
They used "natural born citizen" because "natural born subject" had been used in Blackstone's Commentaries and other legal treatises which had taught them the law. Legal language is like that. Formulas and phrases are passed down through history.
"High crimes and misdemeanors" is another phrase that has a long history and has to be understood in light of that history. It doesn't mean officials can be impeached if they have enough traffic tickets.