My side believes in a Conservative and restrictive view which would successfully curtail abuses and prevent foreign influence in the Executive branch.
Your side advocates a policy of Foolishness, and my side is advocates a policy of common sense.
In short, there's no evidence that "born a citizen" was rejected in favor of "natural born citizen," so no evidence that they don't mean exactly the same thing (no evidence from the Constitutional Convention, at any rate).
My checking says Hamilton Submitted it to the Convention. It is not plausible to believe they didn't read it, and having read it, did not chose to either accept it or amend it. Hamilton was of sufficient social stature to assure his proposals would be considered.
It is possible that they saw Hamilton's version of Article II to be sufficiently similar as to what they already had written so as to make it not worthy of re-writing to include it.
I'll concede the point.
So the question is supposed to be resolved on the basis of results? We should decide what the Framers meant based on what loopholes others may have discovered? It's funny to me that the side arguing that we should interpret the Constitution in whatever way is required to get the results we want claims to be the "conservative" side, while the side arguing that we should accept what the Framers said regardless of how it might be abused is denigrated as the "liberal" side.
The Framers may well have been horrified that a man of Barack Obama's or Ted Cruz's parentage could become president. Unfortunately for you, that doesn't prove they forbade it. They'd probably have been horrified that a man of Mitt Romney's or Eric Cantor's religion could become president, too, but they didn't forbid that either.