I am no constitutional expert but I don’t think that is what it says particularly this part: “of parents not owing allegiance to any foreign sovereignty.”
I believe that the 14 amendment views anyone born in the US as automatically a citizen regardless of the parents status (with one exception for foreign diplomats stationed in the US).
There are two legal principals regarding citizenship that is not gained through naturalization. The first is jus sanguines witch is (the law of blood) which automatically confers full citizenship to anyone born to two US citizens anywhere in the world. The second is jus soli the (law of place) which automatically confers full citizenship on anyone born within the US. These principals have been around for a long time and there is a ton of legal president and jurisprudence which is based on them.
The Supreme Court has never defined the term “Natural Born” because the issue has never come up before. I suspect they would view anyone born in the US or anyone born to two US citizens as a natural born citizen.
Under jus sanguines (the law of blood) the citizenship of the parents is critical to conveying citizenship. Under is jus soli the (law of place) the nationality of the parents is completely irrelevant as we see today with illegal immigrants and their US citizen offspring.
Then they need to rule on it and get this resolved. The anchor baby thing is a whole separate issue that I think needs to be reexamined. If Ahmadinijead's wife is pregnant, he flies her over here, the baby is born on US soil, he/she could be President? That's insane. So either way you look at it, I think SCOTUS needs to step up the the plate. .