It's not just illegal aliens, tourists visiting from Germany or Spain, etc., a woman who happens to be pregnant and gives birth during the visit, they are not "subject to the jurisdiction thereof" so that baby is NOT an American citizen, though he was born on U.S. soil.
There is obviously a great need for the SC to clarify this issue. As you said, it's unlikely that a child born in the US through some unintended misfortune, to foreign tourist parents would be given American citizenship, but I don't know that it wouldn't be so.
There's obviously far too much ambiguity in the law. It needs to be clarified and settled, once and for all.