No, it doesn't, Joseph.
The term came directly and absolutely from the term "natural born subject," which had a long and well-known history, and a very specific meaning.
According to the natural (and divine) law of England, if you were born within a realm, you were "naturally" a part of that realm. It made no difference whether your parents were citizens or legal aliens.
The only exceptions were the children of foreign royalty/ ambassadors, and the children of invading armies.
We adopted the exact same rule. And as the Supreme Court specifically said in 1898, that rule applied in England, in the English Colonies in America, in the United States after Independence, and in the United States after the establishment of our Constitution.
But it's not just the word of the Supreme Court. All of our prior history and law backs that assessment.
As I said, you are advocating for a more foreign U.S. Presidency. I am wholeheartedly opposed to your opinion.