The First Congress made it a priority to explicitly STATE that such children born US citizens abroad were ALSO natural born citizens, and hence eligible to the Presidency.
Yes, they are natural-born citizens, but they are such by statute. According to the State Department's interpretation of the law, statutory NBCs may not necessarily be NBCs for Constitutional purposes. They are not automatically eligible to the presidency.
7 FAM 1131.6-2 Eligibility for PresidencyI believe that SCOTUS would rule in favor of their eligibility.(TL:CON-68; 04-01-1998)
a. It has never been determined definitively by a court whether a person who acquired U.S. citizenship by birth abroad to U.S. citizens is a natural-born citizen within the meaning of Article II of the Constitution and, therefore, eligible for the Presidency.
d.(Skip) In any event, the fact that someone is a natural born citizen pursuant to a statute does not necessarily imply that he or she is such a citizen for Constitutional purposes.
I do too, which is why I often say things like, "Ted Cruz is almost certainly a natural born citizen (in the Constitutional sense) as well."