Actually, you [sic] are merely subject to the judicial laws of the country they invaded. They and their kids don't get amnesty and US citizenship for their kids by breaking our laws.
Jurisdiction has 2 components; sovereignty among nations (citizenship), and legal governance (legislation, judicial) among its people and visitors, within its borders. Treaties fall in between (Executive, Legislative).
You seem to be conflating the matter of citizenship for the alien parents (who can only acquire citizenship by naturalization) and for citizenship for their children if born within the sovereign territory of the United States. Citizenship attained by birth for the children does not therefore mean alien parents automatically become citizens as well (which no one is arguing).
Jurisdiction has 2 components; sovereignty among nations (citizenship), and legal governance (legislation, judicial) among its people and visitors, within its borders.
It flatly does not, insofar as America is concerned. You can see this just be looking at the citizenship clause and how it is worded: "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."
If citizenship is a component of jurisdiction (as you assert), then the 14th Amendment would be nonsensical, because it flatly states that American citizenship is dependent upon jurisdiction. Jurisdiction cannot be dependent upon citizenship if citizenship is likewise dependent upon jurisdiction: it would be self-referential, a circular definition.
Thus, since the 14th Amendment of the Constitution outweighs your opinion, your understanding of jurisdiction is erroneous.