Under the jurisdiction thereof, has to (should) mean more than just being legally obligated to following local laws. Otherwise, what was the point of excluding Indians? They excluded them because they were under the jurisdiction of their local tribes. They excluded them because they didn't want them to be US citizens. The SCOTUS stated in 1831 for Cherokee Nation v. Georgia, 30 U.S. (5 Peters) 1, tribal governments are not US "States" Constitutionally, and they are not "Foreign States," in regard to Article III original jurisdiction.
https://www.fjc.gov/history/timeline/cherokee-nation-v-georgia Jurisdiction applies to Tribal Lands. I imagine if an illegal immigrant was on Tribal Lands they would be under the jurisdiction of that tribe. At that point the The Indian Citizenship Act of 1924 would have to be taken into consideration.
Yet another example of how this jurisdiction argument, in court, goes nowhere. Native Americans are specifically recognized as being under the jurisdiction of their Tribal Court/Laws if you will, and also every single one born is a US Citizen, simply for being born on US soil. Jus Soli again.
An illegal immigrant, who sneaks into the country, has a baby and then claims citizenship for that child, just doesn’t seem to fit with the bigger concept of under the jurisdiction thereof. Except every single court ruling has said they are. Not one case exists that says otherwise.
The Chinese case in the early 1900s that was the foundation for birthright citizenship, seems to, in my opinion, was decided with poor logic. The parents had been in the country for years. Perhaps it’s a case where a certain amount of time qualifies you as being under the jurisdiction of the United States. This would be an argument that uses a logic other than the consistently defeated, as in always, every time, argument of jurisdiction. Time-based evaluation would be an updated, and in my opinion, needed change.