You are correct - Jay was asserting that the "natural born citizen" requirement was the foremost measure that should be taken to prevent foreigners playing a managerial, ministerial, or executive role in the administration of the national government. As I read it, the concern about foreigners led Jay to suggest the "natural born citizen" requirement, as a means of preventing a foreigner from immigrating to the U.S., becoming naturalized, and running for President.
We may disagree about whether a person born in the United States to foreign parents (or, to one U.S. parent and one foreign parent), is a "natural born citizen," but it is quite a stretch to say that such a person is a foreigner. If Jay was concerned about foreigners playing a managerial or executive role in the national government, then prohibiting the U.S.-born child of foreign parents (or, again, one foreign parent and one U.S. parent) seems to be a bit beyond the scope of his concern.
Then why does Article II, section 1, pa. 5 add an exception to the requirement that the President be a "natural born citizen," so that anyone who was a citizen (of any sort) when the Constitution was adopted was eligible to be President? If "natural born citizen" just means "anyone born in the US," THEN THE EXCEPTION IS TOTALLY UNNECESSARY. Read the entire argument Madison makes in which you found that quote to see beyond doubt that that's true.