All of that should serve as proof why the birth certificate is vastly more important than some obscure (and almost certainly incorrect) interpretation of the phrase “natural-born citizen”.
AMEN!!!
I'll even give the Birthers this: their Vatel theory might have held some water, as an explanation of an unclear Constitutional issue, before we had 125 years of actual practice, including electing (Republican) men to the Presidency once, and Vice Presidency three times, who aren't natural born citizens under their theory. The idea that we can ignore all of that, and now advance a different interpretation, all for political expediency, is delusional.
Look at another example. Does the Constitution allow members of the House to resign? It doesn't say. It does specifically allow Senators to resign, but not Representatives. Was that an oversight? Maybe. But some State/Colonial constitutions/charters did not allow House members to resign. Back then and even to this day, British law does not allow Members of Parliament to resign. The thought was that if a man could be compelled to enter military service against his will, he could also be compelled to serve in the legislature.
Now we know that as a matter of practice, several House members resign each term. That unclear issue in the Constitution has been interpreted as allowing resignation.
Suppose Republicans had won the early by-elections, to fill vacancies, in 2009-2010. Then suppose that Speaker Nancy Pelosi, needing votes to pass Obamacare, had said: "Hey, you can't resign from the House, Constitution doesn't allow it, so the Democrats who thought they resigned to take Obama Administration jobs are still members, and they get to vote, not the Republicans who won the by-elections, because there was never a vacancy to fill."
That is exactly what the Birthers are trying to do. Pelosi's resolution of an unclear issue might have been valid in 1790, but not in 2010.