It does envisage the globe trotter from birth. That’s why it specifies that to be eligible the person shall have “been fourteen Years a Resident within the United States.”
Been 14 years... that hardly treats the raised-as-foreigner problem.
“been fourteen Years a Resident within the United States.” That seemingly random number was selected to allow Alexander Hamilton to be eligible. He was a member of the Committee on Style which decided on the wording and, it just so happens, he arrived in the colonies exactly fourteen years before the ratification.