Children born in the US are US citizens. We might prefer that they weren't, but the question isn't, what would we prefer? The question is, what is the law?
"Their parents are subject to the jurisdiction of Mexico. That's why we jettison them back to Mexico."
That's a misunderstanding of "jurisdiction". If they are on US soil they are in our jurisdiction, not Mexico's.
"They are not subject to the laws of our nation other than to be deported or arrested for crimes..."
That's absurd. An illegal alien is subject all of our laws, just like anyone else. They can be charged with crimes just like a citizen or legal resident, and have been many times.
"Mexico screams bloody murder every time we put one of it's favorite sons or daughters on death row. It says we have no right to do that."
That's got nothing to do with jurisdiction. It's got to do with the death penalty.
"Once a foreign national becomes a citizen, they are then "subject to the jurisdiction thereof".
No. As I've said, that's not what jurisdiction means. Jurisdiction means that the laws applies to you. Our laws apply to everyone on our soil, whether one has legal status or not. The only exceptions are those with diplomatic immunity.
https://www.fairus.org/issue/societal-impact/birthright-citizenship
https://fedsoc.org/commentary/publications/birthright-citizenship-two-perspectives
The most you can say is that it has become common to accept the 14th Amendment does say children of illegal immigrants born here are automatically citizens.
Having never been ruled on by the SCOTUS, there has been no determination that you are right. Further, there are good arguments against it as well.
Sorry you didn’t know that.