"All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the state wherein they reside."
is a clarification of Article 2 Section 1's:
"No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President"
Of course, we're used to talking about Article 14 in terms of anchor babies not being citizens (because their parents are subject to the authority of their home country). But it's possible to interpret Article 1's "natural born citizen" text in the same light (qualifications of birth citizenship for president isn't based on where you were born, but if your parents were citizens when you were born).
Frankly, I'd rather win the argument with the Article 14 interpretation, both for prez and for birthright citizenship (no anchor babies), than win the argument over the home soil a candidate was born at.
“because their parents are subject to the authority of their home country”
But that argument is fallacious, since we charge such illegal immigrants with crimes all the time. We couldn’t do that if they were not subject to the jurisdiction of the United States the second they crossed the border.
People who have children here but are not subject to the jurisdiction of the United States are foreigners on diplomatic missions, not illegal immigrants.
"All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the state wherein they reside."
is a clarification of Article 2 Section 1's:
The 14th amendment is a naturalization law. It cannot make natural citizens by law.