All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside.
That establishes citizenship at birth. Congress can grant extra situations in which someone is granted citizenship at birth, but the Constitution does indeed have a say in this issue.
Congress is given full authority to determine naturalization procedure and law, but the only two forms of citizenship that is contemplated by the Constitution, according to the Supreme Court, are citizenship by birth (whether given by the Constitution or by laws of Congress) and citizenship by naturalization.
There is not a third form of citizenship.
This is a tighter requirement than simply citizen or naturalized citizen, just like citizen at least 35 years old is a tighter requiremeet than just citizen. So, natural born is an understood requirement for office, not a Constitutional definition of who is a citizen.
What if Article II were instead written as:
"No person except a Citizen, natural born, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty-five Years, and been fourteen Years a Resident within the United States."
We don't argue that "citizen over 35" or "14-year resident citizen" are additional forms of citizenship, so why the insistence that "natural born citizen" IS a form of citizenship to be rejected, when the context is clearly to further qualify what type of citizen is eligible to be President?
-PJ