The founders considered it an issue of National Security. See the Federalist Papers number 68. They put the distinction in because the best way to assure the loyalty of a person is to know that they are solely a native of the nation and of no other. The founders did not wan a Commander in Chief of the armed forces who could be potentially compromised.
And yes, if we fail to uphold that distinction then the son of Osama Bin Laden and an American woman could potentially be President of the United States.
And I'm not disputing that. Perhaps a question is the best way to explain:
If Hugo Chavez declared Sarah Palin as a natural born citizen of Venezuela, with all the rights and responsibilities of that same citizenship, would she be barred from seeking the Presidency?
Answer that question, and you have answered the issue with Arthur and Obama in terms of their legal status regarding election to the Presidency.
Brilliant post here.