Not at all. I am perfectly content with it being parentage alone. (The System Most of the World Uses. Even England now uses it. ) One's parents can always be determined by an examination of the child, but one's place of birth? Not unless you already know it.
If you lack that information, you cannot tell if a child is an American or not. A rather silly conundrum, don't you think?
And no, parentage is not always able to be determined any more than birth place. You find an orphan wandering the streets of NYC - who are his parents?