You said that stepping foot on US soil makes them US persons. It does not. As you said above, they can claim asylum after illegally entering, but even that is subject to legal interpretation.
I've read here today, that aliens seeking asylum are supposed to do so at the American consulate in their own country, not at our border.
Bottom line - the President has it within his power to suspend all grants of asylum for those doing so at our border.
Ok you know more than they just briefed us by the border patrol. I was just briefed on this very topic, but I’ll defer to the Internet genius.