It will require an amendment to your highly distorted and incorrect understanding of the meaning of “and subject to the jurisdiction thereof.”
That formulation is not and never was a reference to the so-called exceptions to jus solis, nor has the law of citizenship of the United States ever been exclusively that of English common law, although I will grant that the english common law tradition carried more weight in some States than in others. I will remind you that even after the founding, it was the STATES, not the federal government who determined their own citizenship, and most were more than just a little bit motivated to distinguish “citizenship” from “subject-hood.” Citizens are those who are tied to their country by ties of land and blood. They are not mere whelps of accidental or intentional invaders, and that is precisely what children of those here without permission or authority are.
In reaction to this and abuses by States, 14A was passed and ratified. It permanently took away the power of states to determine citizenship.
As for "subject to the jurisdiction thereof," it needs to be looked at for their time, not ours. There were no illegal aliens at the time, because there were no restrictions on immigration.
Jurisdiction: the right, power, or authority to administer justice by hearing and determining controversies.
Illegal aliens are subject to this in our countries, though it is sadly often unenforced. Legal armed invaders and those with diplomatic immunity are not.
By your definition, we have no right to prosecute or imprison those here illegally. They are not "subject to the jurisdiction thereof.