There was one other group excluded by the "subject to the jurisdiction" language: Native Americans--not all Native people, but those who remained on reservations and were "subject to the jurisdiction" of their tribal governments. By law, they could not be sued in federal or state courts, or arrested and held by local authorities. Disputes with these Natives were handled as government-to-government matters.
Again, "illegal aliens" don't fit this model; they can, as noted above, be sued, arrested, jailed, and even executed by U.S. authorities for crimes they commit here; and their children can as well.
The second strand of the argument runs that the authors of the Clause couldn't have meant what they said, because there weren't any "illegal aliens" back then, or even all that many immigrants at all. Coulter again: "Inasmuch as America was not the massive welfare state operating as a magnet for malingerers, frauds and cheats that it is today, it's amazing the drafters even considered the amendment's effect on the children of aliens."
In fact, America in 1866 was much like America today. (Anybody who wants my detailed look at the situation at that time should feel free to consult my book, Democracy Reborn, which tells the story of the Fourteenth Amendment.) Roughly the same proportion of the population--quite high, by historical standards--was foreign-born in 1866 as it is today. Many Coulterish voices in the press and the pulpit foresaw the end of America. Of course, the undesirable groups were different--beery Germans, wild Irish, and sinister Chinese rather than insinuating Latinos--but the perceived menace to the "real" America was just as dire.
Andrew Johnson and his pro-Southern allies warned that the language would make citizens even of Gypsy children--who, one opponent warned, "pay no taxes; who never perform military service; who do nothing, in fact, which becomes the citizen, and perform none of the duties which devolve upon him, but, on the other hand, have no homes, pretend to own no land, live nowhere, settle as trespassers wherever they go."
Yes, the amendment's authors replied: Chinese, Irish, and Gypsies, too. The point was to create a nation in which all children began as equals, where local majorities had no power to deprive them of their basic rights.
Make no mistake: that idea, of a republic of equal rights and rule of law, is the real target of the citizenship-restriction movement. Repeal of birthright citizenship would create a new class of American untouchables. That's all the people of Arizona, or Prince William County, Virginia, or Hazleton, Pennsylvania are asking--a small hole in the legal idea of equality to let them push around one group they really, really dislike. That group and their children, that is; and, for that matter, their children's children, even unto the last generation.