But in that speech Madison made before Congress, he argues that everyone born in the territory of the States that formed the US were citizens of the US by place of birth, and so eligible to serve in Congress.
So the reason the exception was needed is because no one born before the country was formed had parents who were citizens of the US at their time of birth.
Again, you are refuted by your own reference.
This permission of a naturalized citizen to become president is an exception from the great fundamental policy of all governments, to exclude foreign influence from their executive councils and duties. It was doubtless introduced (for it has now become by lapse of time merely nominal, and will soon become wholly extinct) out of respect to those distinguished revolutionary patriots, who were born in a foreign land, and yet had entitled themselves to high honours in their adopted country. A positive exclusion of them from the office would have been unjust to their merits, and painful to their sensibilities
Regarding the natural-born citizen requirement itself, Storey writes:
But the general propriety of the exclusion of foreigners, in common cases, will scarcely be doubted by any sound statesman. It cuts off all chances for ambitious foreigners, who might otherwise be intriguing for the office