Citizen can be based on whatever definition you choose, place of birth alone if you wish. But after passing the citizen test, one must pass the natural-born test to be president.
You haven't addressed why the Founders added the additional qualifier of natural-born (and excluded it in the grandfather clause) if place-of-birth is enough to define ordinary citizenship.
-PJ
I shouldn't need to.
A person born offshore might become a statutory citizen under ordinary citizenship statutes which Congress has the power to adopt; not naturalized but a citizen at birth under the circumstances of birth--in fact we have a bunch of statutes like that.
That person shouldn't be natural born and shouldn't qualify to become President because the person is subject to the sovereignty of the head of state of the place of birth which might override his decisions as President.
We could (but do not) have a statute that says a person who resides here twenty-five years becomes a citizen on the twenty-fifth anniversary. Same answer; same reason.
Naturalized citizens shouldn't be eligible either--same reason.