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.
Far better to get all the paperwork in order as soon after a child's birth than to try to reconstruct events and chase down documents and "witnesses" 40 or more years later -- as is happening now.
If from day one, all the documentation clearly establishes that a child has been legally determined to be either natural born or native born, then there shouldn't be any disputes later on.
It shouldn't add that much more expense. The parents present their birth certificates, which state their place of birth, at the same time that the child's birth certificate is prepared. The legal/recording clerk/registrar then checks one of two citizenship boxes on the child's birth certificate, and the parents' birth certificates and child's birth certificate are kept on file.
Hassle-free. Plus it establishes a clear chain of document custody since the time of a child's birth.
I have the feeling at some point both Congress and the SCOTUS will be nailing down the legal definition of natural born in the future. If Congress and the Court determine that there is a distinction between natural born and native born, then these determinations will be made at the time a birth certificate is prepared and legally filed.