What matters is the text of the Constitution, not simply intent. We can use intent to help understand the text as written. We can also use operative historical meanings in parallel contexts when we do not know what a word or phrase means. However, Mr Madison in your passage does not use the term “natural born”. He may use it elsewhere, but you did not cite it. Do you have such a reference?
We do have the text of the Constitution drawing a distinction between “natural-born” and all other “citizens at the time of the adoption of this constitution”. The latter would include native born (not natural born) and others naturalized by processes in effect then. All the citizens of the latter class “alive at the adoption” are dead. That leaves the remaining class — natural-born — as those only eligible.
Very few, if any, qualified legal authorities agree with you.
Why is that?
Qualified legal authorities will tell you that “Native Born” and “Natural Born” mean the same thing, and that it was not necessary to distinguish “distinctions without a difference”.