If the US government has in fact legally established three types of citizenship, only one of which enables a person to serve as president/vice president, you'd think that (excluding naturalized citizens) we'd be classified as natural born citizens or just plain citizens at or near the time of our birth and that our government-determined citizenship classification would appear at least on our birth certificates, our passports, our census forms, and maybe our death certificates. That's the kind of stuff the government normally likes to keep track of.
While it definitely wants us to be legally designated male or female, the US government has never bothered to legally establish which of us is natural born. That's very curious. I think it can be argued the US government's curious indifference means that it does not consider natural born a legally distinct category of citizenship.
To what purpose?
The Rights of citizens remain the same, whether Natural Born, native born, or naturalized.
To create a 'second class citizen' status flies in the face of equal protection under the law.
The only time being a Natural Born Citizen matters is when running for President or Vice President, or otherwise ascending to the office (as in the unprecedented instance of the President and Vice President suffering their demise at the same time).
As such, as a qualification for the office, it can be sorted out when it arises in those cases, rather than be done for everyone.
It would just be an added expense.