What better way to seek to ensure this quality in a President, than to require that he (or she) be of The People, himself (or herself)?
Is the child of a foreign citizen really of The People? Allegiances will have formed elsewhere, for such a child. Empathy will be lacking, for what Madison termed "the great Mass of the people."
So, the Framers sought to preclude such individuals, from ever attaining the highest office of the land, for this reason. They were prescient in their concern. We're now living the consequences of having ignored them.
If you look at the Constitution’s requirements of the president, one stands out: that he shall take good care that the laws be faithfully executed.
Let’s start with his oath of office, where he swears to preserve, protect and defend the Constitution. = FAIL.