Laurence Tribe points out that Ted Cruz' own originalist standard for constitutional interpretation (something Tribe should know about since he was Cruz' constitutional law professor) would deny Cruz the presidency.
Growing up, I always understood that you had to be born in the United States, not simply born to American citizens. The constitutional provision had disqualified Alexander Hamilton from being president, for instance (and it seems that I read once that some thought this was intentional). I dated a girl in college who was very political, but because she had been born of missionary parents in the Philippines, she had always been told she couldn't run for president.
I wondered about John McCain, who kept saying he was okay because the Panama Canal Zone was U. S. Territory at the time.
I think the provision is antiquated and ought to be changed, as does Tribe. But, as Tribe points out, this is the standard of those of us who believe in a living constitution, which Cruz does not.