They had no problem with foreign influence. The President has to be 35 years old, but only has to be a resident of the U.S. for 14 years. So the someone who was born a U.S. citizen but raised in a foreign country from ages 1-21 could return to the U.S. and be elected President 14 years later.
Permit me to hint whether it would not be wise and seasonable to provide a strong check to the admission of foreigners into the administration of our national government
If one is born with foreign citizenship, and thus owed allegiance to a foreign country, one was considered a foreigner. Citizenship, in 1787, followed that of the father (for children) and the husband (for wives).
Their inclusion of the "natural born Citizen" clause had nothing to do with the (yet to be defined) national laws on naturalization.