"subject to the jurisdiction thereof"
This was written to deal with native americans. Not Mexican nationals.
Actually it was written to deal with native americans. The "subject to the jurisdiction of" clause was intended (and congressional debate proves this) to exempt: Indians subject to tribal law, diplomats (protected by diplomatic immunity), and persons of other countries under American military control. But since the ratification of the 14th Amendment, all children born in the United States to immigrants have been citizens. You don't have to like that from a policy standpoint, but it's settled law.