“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.
I think you are confused.
Critics erroneously believe that anyone present in the United States has “subjected” himself “to the jurisdiction” of the United States, which would extend citizenship to the children of tourists, diplomats, and illegal aliens alike.
But that is not what that qualifying phrase means. Its original meaning refers to the political allegiance of an individual and the jurisdiction that a foreign government has over that individual.
The fact that a tourist or illegal alien is subject to our laws and our courts if they violate our laws does not place them within the political “jurisdiction” of the United States as that phrase was defined by the framers of the 14th Amendment.
This amendment’s language was derived from the 1866 Civil Rights Act, which provided that “[a]ll persons born in the United States, and not subject to any foreign power” would be considered citizens.
IMHO look at decades preceding the 14th Amendment. For example, one reason we went to war with England in the War of 1812 is because England would capture American sailors (some say a total of 10K within 10 years) and force them to work on British ships (impressment). To be fair, some of the "Americans" at that time were Brits who had left England to come to America, never became naturalized citizens, and, therefore, were subject to the English draft laws as any other Brit. Thus, those men weren't "subject to the jurisdiction" of the U.S. but instead were subject to the jurisdiction of England. In other words, those men had to obey England even though they were in the U.S. (I'm not talking about the American citizens the Brits impressed because they looked like Brits. I'm talking about the British citizens the British navy impressed because they had to serve in the military like other men.)
That doesn't mean that immigrants back then didn't have to obey our laws regarding theft or murder or such. It just means that there are some laws that citizens have to obey of their country of citizenship regardless of where they live (at least back in the 19th century when the 14th amendment was written). IMHO, that's what "subject to the jurisdiction of" means. It means, which nation can draft you into their army (or other things a citizen is required to to) or which nation you can vote for, etc.
If their governments can demand they return, then they are still subject to foreign jurisdiction.