The Supreme Court has ruled that every word in the Constitution has meaning.
Natural born citizen is three words. Adding ‘natural’ requires one must be something more than a born citizen. The Supreme Court has also said that the term "natural born citizen" is simply an adaptation to our form of government of the English term "natural born subject." And in the opinion where that is analyzed at length, the term is sometimes written "natural born citizen" and sometimes "natural-born citizen." The hyphen doesn't affect the meaning.
The Supreme Court has also said that the term "natural born citizen" is simply an adaptation to our form of government of the English term "natural born subject." And Thomas Jefferson says they're wrong. He changed the word. It is silly to think he changed the word without intending to change the meaning.
He got the usage of the word from Vattel. Had he intended us to follow English law in defining the character as that of a "Subject" he would have used the word "Subject." He almost did, and then he remembered that we were following Vattel's natural law basis for Independence.
We got the Idea of Independence from Vattel. We didn't get it from "English Common Law", because Independence is strictly forbidden to "subjects."