So if I go overseas and father a child, I can bring my child home with no paperwork and that child could be the president someday? How many people are walking around in Vietnam right now that fit that description?
That’s the same logic you just used.
The paperwork was a birth certificate showing the mother to be a US citizen.
You left out the MARRIED part of the equation.
And, YES!
The big thing that keeps this from happening, doubt those American soldiers went to the U.S. Consulate and registered the birth and received a CRBA and passport for these children.
+2.