It's not a non sequitur. The argument was that foreign-born citizens might have conflicts of interest and should therefore be barred from the Presidency, and it logically follows that other citizens who might have conflicts of interest should also be barred from the Presidency. It's a reductio ad absurdum: if one believes that the potential conflict should not be a bar to Catholics, Jews, Hispanics, or other groups that might be conflicted, it should be no bar to the foreign-born either.
George W. Bush was born in New Haven, CT. Had he been born a few miles to the north, across the Canadian border, and moved to this country as a baby, would he be a worse President?
I do not support this Amendment so that Schwarzenegger can be President... nor, for that matter, for the sake of Albright, Soros, or Granholm. I support this Amendment because nobody has control over the accident of his birth, and some of the most fiercely patriotic Americans are immigrants.
It IS a non sequitur. The founders and the documents assuned that loyalty to God does not conflict with loyalty to the nation. Loyalty to another nation would so conflict. Liberals and other sorts of socialists assume that loyalty to anything other than the state at any level conflicts with loyalty to the State. Conservatives and Constitutionalists take the Constitution at its word.