“So can foreigners, so what’s your point?”
Foreigners who are within the borders of the U.S. can be brought before a Federal judge, can compel witnesses for their defense, have benefit of the Grand Jury if charged with a capital or infamous offense, are protected against unreasonable searches and seizures, etc, etc.
If that’s not being subject to the jurisdiction of the United States (and re Supreme Court has ruled repeatedly that it is), words have no meaning.
And like illegals they are not subject to “full and complete jurisdiction on the part of the United States”, they can not participate in civil society, ie vote or hold federal office.