As a side note, “could” and “should” are two different categories. As Obama and Biden have shown us, anyone “could” break laws and become President or Vice President, but they should not have done so.

Here were the authors’ intentions behind the 14th Amendment::

John Bingham | A Matter of Allegiance

"To naturalize a person is to admit him to citizenship. Who are natural-born citizens but those born within the Republic? Those born within the Republic, whether black or white, are citizens by birth -- natural-born citizens. There is no such thing as white in your Constitution. Citizenship, therefore, does not depend upon complexion any more than it depends upon the rights of election or of office. All from other lands, who by the terms of [congressional] laws and a compliance with their provisions become naturalized, are adopted citizens of the United States; all other persons born within the Republic, of parents owing allegiance to no other sovereignty, are natural born citizens. Gentleman can find no exception to this statement touching natural-born citizens except what is said in the Constitution relating to Indians." [Congressional Globe, House of Representatives, 37th Congress, 2nd Session] https://memory.loc.gov/cgi-bin/ampage?collId=llcg&fileName=059/llcg059.db&recNum=680

“Every human being born within the jurisdiction of the United States of parents not owing allegiance to any foreign sovereignty is, in the language of your Constitution itself, a natural born citizen.” [Congressional Globe, House of Representatives, 39th Congress, 1st Session] https://memory.loc.gov/cgi-bin/ampage?collId=llcg&fileName=071/llcg071.db&recNum=332