That was not the intent of the drafters of the 14th Amendment. It was directed primarily at slaves who though not Citizens, were subjects. The 14th Amendment made all children who were subjects of the United States at the time of their birth Citizens. Visitors who are on Visas from foreign countries, or invaders (as in illegal aliens) are not subjects of the United States, but are subjects of the country of their origin. Becoming a subject of the United States requires an intent to stay permanently and to subject yourself to the sole jurisdiction of the United States.
I do not believe that anchor babies are legitimately Natural Born Citizens or even Native Born Citizens (if there is still a distinction after the 14th Amendment) because their parents are not legitimately subjects of the United States and are legally, in all sense of the term, subjects of the country of their origin.
What Natural Born Citizenship is intended to convey is a singular loyalty and anyone possessing dual citizenship at the time of birth is not a Natural Born Citizen of the United States unless the parents are Citizens, which would make the child born on American soil solely an American Citizen at the time of birth.
So I vehemently disagree with your analysis of the effect of the 14th Amendment. You have to go back to the original intent of the drafters of the 14th Amendment and if you do, you must conclude that the children of visitors and invaders were never intended to be granted Natural Born Status.
I agree, but I also agree with Scalia that the intent of the drafters is not what determines the meaning of a law or constitutional provision. The drafters, as such, have no constitutional authority or status at all. If they have such an intent, then it is there responsibility to put that intent into writing so that the people who are tasked with enacting it into law can see the meaning on its face.
The folks who turn a piece of paper into a law/constitutional amendment are the members who pass it, and even more importantly, the states that ratify it. So, you go by the ordinary meaning of the words as they would have been understood at the time by the people who voted for/ratified it.
The ordinary meaning of the words used is not limited just to former slaves, as it makes no mention of them. The best argument I've heard about "subject to the jurisdiction thereof" is that it excused Indians living on tribal reservations not fully subject to U.S. law. Perhaps you could stretch that to folks who are here illegally as well. But if the 14th was intended only to apply to freed slaves, why didn't they simply say that expressly in the Amendment itself rather than drafting it so broadly?