In spite of the baldarah..
I am not sure the issue of Obama's birth place was as much about Constitutional legality as it was about his wanting to be seen as being born in the USA.
In other words, had Obama acknowledged that he was born in Kenya, he would still qualify but since no foreign born person has ever been POTUS, it would have been a political liability (at least equal to being Catholic) and the question of why he had originally claimed to have been born in the USA would have been a talking point.