Actually, no.
Under the laws in place at the time, a child born of a Bumfuqistani in Bumfuqistan is a Bumfuqistani.
Think that one through, if a single American parent qualified a whelp to be an American, every kid born within 100 km of an overseas US base would be claiming that daddy was a US Sailor (named joe).
Further, merely being a citizen does not qualify one to be president.
The Founding Fathers reserved that for a very special status, one that appears nowhere else in American law: That of NATURAL BORN citizen.
They wanted the President to be free of any hint of divided loyalties. (Imagine McCain was elected, and the mess in Honduras was in Panama. Would any decision he made with regards to a Panamanian constitutional crisis be subject to second guessing because of his birth ties? You bet it would!)
In my opinion, and in agreement with the source documents the Founding Fathers were using (Vattel and Blackstone) a natural born citizen was clearly, unquestionably and obviously only someone who was born within a country, of that country's citizenS.
CitizenS. Plural. As in BOTH parents.
As a U.S. citizen, his mother could have at least had him naturalized upon return to the States, something the Mexican woman in your example couldn't have done. That is all I'm saying.