This is a common fallacy. Being born in a barn will not make you a horse. Likewise, being born in the United States does not make you an American. This mistaken notion has been repeated endlessly since the 14th amendment because simple minded people cannot understand the distinction between "Citizen" and "Natural Citizen."
The difference is whether another nation has a legitimate claim on your loyalty. A "natural citizen" is someone who cannot be regarded as a citizen of any other country under any application of international law. The principle of law known as Jus Sanguinus is recognized among many countries, and is indeed the sole basis for citizenship in those countries.
Here in the United States, we have had an endless parade of exceptions to the "born here makes you a citizen" theory. First of all, Indians who were born here were not citizens. Slaves who were born here were also not citizens. Colonists who were born here, yet chose to remain loyal to the British Crown during the Revolutionary war (Called Loyalists) were also not American citizens. Women who were born here but married foreign citizens were also not citizens. (Till the Cable act of 1924) And lastly, any child born here from the staff of a foreign embassy was NOT an American citizen.
That is 5 (five) exceptions to the theory that being born here makes you a "natural citizen." I would point out that it took an act of Congress to change the citizenship status of Indians, Slaves, and Women. How could the status of "natural citizen" be dependent upon a subsequent legislative act of congress? If it requires a special law to make it occur, how is it "natural"? Obviously in 1787 they didn't have such a law, so those who are citizens only on the basis of a subsequent law cannot be those "natural citizens" of which the U.S. Constitution speaks. They could not have existed prior to the law which makes their citizenship possible.
One last thing. James Madison (The father of the U.S. Constitution) wrote under the pen name Publius. Here is an excerpt from a letter he sent to two Virginia newspapers in 1811. In this letter he full out says that being born here may make you a citizen of a state, but it WILL NOT make you a citizen of the United States.
Here is a link to the ENTIRE page of that Newspaper where you can see the entire letter. (Scroll down to page two.)