That's what matters. The 14th has nothing, zero, to do with who is a "natural born Citizen."
Period.
What matters is that natural born subject was a very well known and very commonly used legal phrase in the colonies, and it was used interchangeably with natural born citizen for some years after the Constitution.
And that legal meaning included the children of aliens, unless the aliens were ambassadors or in a foreign army.
NBC was NOT an appeal to European ‘natural law’, but to a term used regularly by every legislature that ratified the Constitution. They didn’t need to define it because it was already a well known and defined term.