What the framers intended (and they said so) was to prevent the British from running candidates, so to keep the Crown from undo-ing the revolution at the ballot box. So, they limited the POTUS (curiously, no other positions) to be men that were either already here when the Constitution was ratified, or were born here. It was explicitly intended to keep out the Brits. But it also serves to keep out those others that might have divided loyalties.
Yes, that is right.