No, it is not. If "natural born citizen" means "anyone born in the US," then the exception was TOTALLY NECESSARY, because, in 1789, there was NOT A SINGLE PERSON who was 35 years of age AND had been "born in the US," because the US did not exist 35 years prior to 1789.
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.