Interesting that this issue has once again popped up. This linked article goes into great detail on exactly
what "...subject to the jurisdiction thereof..." means. The common interpretation, which I believe is accurate, is (paraphrase) "those individuals born in-country (jas solis) who are bound by the laws of the land both territorial (city, county, state) and nationally (federal)" are citizens. The question then becomes what binds an individual born in this country to the law? Based on the context of the linked article, it's more than just being born here. Even the article makes it clear that atleast one parent must be a U.S. citizen (born here or naturalized) in order for U.S. citizenship to be conferred onto the child. Unfortunately, the ramifications of
Wong Kim Ark kinda muddied the waters on that simple prerequisite. Our INA laws have since allowed for infants born in the U.S. to non-U.S. citizen parents to be granted U.S. citizenship based solely on jus soli which I belief is a mistake. (See INA, Sect. 1401(a) and/or U.S. Foreign Affairs Manual, Ch 7, Sect 1111(a) & (b)).
While I agree that granting blanket U.S. citizenship to infants born in the U.S. to non-U.S. citizen parents is wrong, I do belief that the question of granting a child U.S. citizenship and subsequently declarating that child's citizenship status to be "natural-born" could and should be more clearly defined because it still leaves open the question of "dual-citizenship" falling under the "natural-born" status when only one parent is him or herself a U.S. citizen, such as the case with our President.* Granted, the latter isn't the issue here, but I'm sure someone will bring it up again.
Bottom line: IMO, if neither parent are U.S. citizens, the child born unto illegal alien parents shouldn't be granted U.S. citizenship. To those who say the child shouldn't be issued a birth certificate but instead be issue some other form of birth document, i.e., an affidavit of birth born abroad, I agree. This would help eliminate the automatic declaration of those children born here to non-citizen parents who themselves have no intention of becoming U.S. citizens. (See Wong Kim Ark above as the exception and not the rule here.) Of course, I think it would take changing the 14th Amendment and/or INA laws to further clarify this issue.
*The issue of Pres. Obama's birth has been upheld in lower district courts where the question of a child's citizenship and natural-born status has been upheld by virtue of said child being born to one alien parent and one U.S. citizen parent.