Every country on the face of this earth claims to have jurisdiction over everyone within it’s physical territory excepting only those that have been granted diplomatic immunity.
If a US citizen goes to Italy and is suspected in a murder case, the legal process will unfold according to Italy’s laws, not the US laws. Thus the US citizen is subject to the jurisdiction of the legal system of Italy while on Italy soil. The reverse is also true. An Italian citizen comes to the US is subject to the jurisdictions of US laws (Congress) and the US legal system (courts) while here, legally or not.
Ever hear of “Indians?” Ever hear of “Indian casinos?” Why have those casinos been able to operate operate “against the law?” Simple - the “Indians” were born in the United States, but they are not “subject to the jurisdiction thereof.”In a similar way, foreign nationals (from Mexico or anywhere else) who enter the US against the law, and who are legally required to turn themselves in and leave America rather than colonizing it, are not “subject to the laws thereof.” They do not subject themselves to the laws of the US. Their very presence is illegal - they cannot be in the country and simultaneously claim to be “subject to the jurisdiction” of the US. That is inherently untrue.
At most it requires a mere act of Congress - not a constitutional amendment - to clarify that the government of the US does not accept the colonizing the US.