In ordinary times, the notion of establishing precedent that immigrants are not under the jurisdiction of the US would be met with sputtering indignance for all but the most idealistic anarchists. Because that goes way beyond open borders, it promotes foreigners to super-citizens untouchable by the law.
First of all, if an undocumented person commits a crime, they're not going to just let them go because they didn't have a birth certificate or passport or a documented name. They'll charge you, try you, and convict you same as anyone else. That's what it means to be subject to the jurisdiction.
The "subject to the jurisdiction" clause has two reasons. First and foremost is American Indians. American Indians, at the time, were explicitly not subject to the jurisdiction of the United States. Various tribes had treaties with the US government explicitly exempting their tribe members from federal, state, and local jurisdiction. At the same time, they were not considered US citizens. An American Indian could literally commit murder and the only legal recourse was to petition their tribe to bring justice to them. (there was also extra-legal recourse, which was a lot more common, but that's neither here nor there)
The other reason is diplomats. Diplomats have diplomatic immunity. They commit a murder? Not a lot you can do. (Hollywood overstates this, but not by as much as you might think) They didn't want to grant citizenship to the child if every diplomat.
This doesn't apply to illegal immigrants. There is no treaty or existing legal framework which grants them immunity to the law.