During the original debate over the amendment, Senator Jacob Howard of Ohio, the author of the citizenship clause described the clause as . . . excluding not only Indians but persons born in the United States who are foreigners, aliens, [or] who belong to the families of ambassadors or foreign ministers.
Why is it we still don't allow children born to Ambassadors to be citizens, under your description they would be dual citizens no matter what.
There is a huge distinction, however, between immigrants (legal and illegal) and Ambassadors. Ambassadors (and, to some extent, their families) often have diplomatic immunity - that is, they cannot be prosecuted for crimes committed in the U.S., and cannot otherwise be held liable by the U.S. for their conduct while here (though they can be expelled). They are the clearest example of people who are here, but who are most certainly not subject to the jurisdiction of the United States.
So, while it is arguable whether or not aliens are subject to the jurisdiction of the U.S., it is clear that Ambassadors are not - hence, it makes sense (at some level) to treat the situations differently.