There is a broader political question that needs to be considered. The loyalty of the parents to the United States. Obamas father was an alien and his mother all but alienated from her country. Obamas loyalty to this country can be questioned by the lack of any deep emotional ties. The woman he married certainly doesnt show any deeper loyalty than his mother did. By contrast, Cruzs father and mother both have lives deeply rooted in this country, as do the immigrant parents of the sentry from Florida. In using the word natural the writers of the Constitution were deeply interested that no candidate have conflicting political loyalties. It was wide suspicion about the loyalty of Catholics to the papacy that kept anyone of that faith from being elected president until 1960.
You make a good point about the founders' intent to promote candidates with unambiguous loyalty to the United States, but at this point I think we have to rely on the voters to sort that out, because the "natural born" requirement won't help. I hope the voters are up to the job.