"Section 1. 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."
Those in the USA illegally are specifically evading the jurisdiction of the federal government.
This was the consistent interpretation and practice all the way into the 1960s.
The alternate argument is that anyone in the US is "subject to the jurisdiction" of the US unless somehow immune, eg, a diplomat. I don't see it as very likely that federal courts today would agree with your argument. Nevertheless, this is the argument I've heard offered in support of the constitutionality of denying citizenship for US-born children of undocumented immigrants. Presumably there is some caselaw to support this? Please feel free to educate me with a citation here.
Those in the USA illegally are specifically evading the jurisdiction of the federal government.
This was the consistent interpretation and practice all the way into the 1960s.