Was there a fear that some silver tongued Englishman would trick the Americans into giving up their independence? Sleeper agents infiltrating federal politics from Europe? Or would it have been understood as barring someone born a slave, or a Native American?
Is there any particular reason it applies only to the presidency and no other office?
It was a reaction against the takeover of the crown of Britain by the Hanovers. It was bad enough to have a leader across the Channel with his real interest in his home country he had spent 54 years in, speaking English as a third or fourth language. Given the distance across the Atlantic, and the rate at which news travelled then, it would be a nightmare and threat to the balance of powers to have the leader's ministers and Congress dominant over national proceedings while some rich aristocrat bought an election for the prestige and remained in their European holdings. (hence why it's next to the 14 years residency part, and also partly why the titles of nobility stuff is in Article I)