Did you read what you cited?
“...and therefore every child born in England of alien parents was a natural-born subject, unless the child of an ambassador or other diplomatic agent of a foreign state, or of an alien enemy in hostile occupation of the place where the child was born.”
You then write “The child of an illegal alien would NOT be a NBC because they were not under the authority of the government at the time - just as an ambassador or foreign soldier is not.”
Illegal aliens are subject to our laws the moment they cross our border. Ambassadors are not, and foreign soldiers are subject to the laws of war in the situation described (”hostile occupation”).
Illegal aliens are subject to American jurisdiction so long as they are within our country. In a conflict between our laws and the those of their homeland, ours would trump their homeland every time (selective enforcement and selective prosecution notwithstanding).
“Illegal aliens are subject to our laws the moment they cross our border.”
ILLEGAL aliens are, by definition, here IN OPPOSITION to our government! ILLEGALLY! They are the equivalent of a foreign army.