Actually, in the case of Obama, it was his birth certificate, or COB to be exact, that was at issue.
This affected his claim that he was born in Hawaii.
IMO, the natural born issue was not really relevant unless one could prove he was not born in Hawaii. If the Hawaii case had born fruit, then it might have gone to a natural born issue, but that would have failed as the mother was a US citizen. But they may have argued that the father did not spend enough time in the US...but I don’t really know..
It is interesting to note that Breitbart just issued a breaking story that said the the 2008 Hilary for President Campaign started the Birther Movement against Obama:
This tells me that she won’t hesitate in doing it again against Cruz.