And what about my other point/question - is the child of a non-citizen, born in the U.S., a citizen at all?
I believe it was customary at the founding, to allow the state the child was born in, to say so, or not. I've recently read that James Madison complained about that (he complained a lot, but I agree).
That is the "anchor baby" dilemma that remains a point of contention. Under current law, he's a Citizen. He, however was not a "natural born Citizen," to the framers, by all we know about them.
They used those two modifying words for good reason. They were our highly educated founders/framers and they were writing our Constitution, after all.
And why use extra words, anyway? The sheet of vellum is only so big.