Jurisdiction in this case does not just mean subject to the laws, or where one resides, it also means allegiance. It means that the child (via the parents) shall have no allegiance to another country (in this case, Mexico). The Fourteenth has been misinterpreted in this manner for far too long. It's been around for more than 100 years, but it's only been within the last couple of generations where an anchor baby has been granted citizenship.
Sorry, but that's a novel interpretation of "jurisdiction" and not one that the courts have ever agreed with. I don't see why we should accept it just in this case when it happens to support what we want.
As to intent, the quotes from the legislators in the link you provided don't support the author's contention at all. They are about children of ambassadors and children born into Indian tribes which have some degree of national sovereignty - both special cases.
Finally, it's very risky to selectively pick some legislator's comment made during debate and then claim that it determines the intent of the legislation (or amendment). I'm certain that with some research I could find comments made by other legislators at the time that would contradict.
That's why we have to rely on what's written, and like it or not the 14th is pretty clear.