I always find it funny when people claim they knew the founders wanted no foreign influences. The founders did not want a women to be president either. The founders did not think blacks were human.
But to your point, a lot of constitutional scholars believe the Natural Born Clause was added to make eligible American Citizens born aboard. Just the opposite of the prevailing thought.
Personally I believe that Cruz is eligible and your opinion might be different. So, we agree to disagree.
All of these Presidents were born under foreign influences (until 1841).
George Washington
John Adams
Thomas Jefferson
James Madison
James Monroe
John Quincy Adams
Andrew Jackson
Andrew Jackson
William Henry Harrison
a citizen at the time of ... , covers all those until there could be natural born ciizens