But as djf noted, de Vattel was Swiss. The Swiss did not and do not have jus soli; to be Swiss, you either have to naturalize or have Swiss parentage.
Who had jus soli at the time of the Founding? The United Kingdom. As Justice Gray wrote in Wong Kim Ark,
As you noticed, more important is that United Kingdom had a phrase similar to "natural-born citizen": "natural-born subject."
From Wong Kim Ark,
"Natural-born subject" appears no less than twenty-two times in the Opinion of the Court for United States v. Wong Kim Ark. We're not talking about a translation of four French words here; we're talking about a phrase in ENGLISH. You really think the founders, who until 1783 were considered natural-born subjects of the British Crown, would use "les naturels ou indigènes" to define "natural-born citizen" over an equivalent English phrase, "natural-born subject"? Or that they'd use de Vattel's definition over the English common law definition? If the Founders had wanted to use de Vattel's definition over the English common law definition, they would've made sure to emphasize that they picked a definition contrary to English common law. They did not.
It wasn't until 1983 that the United Kingdom revised its jus soli policy.
Some of you are dissatisfied with our current nationality law. Evidently, some of you would like a law closer to what the United Kingdom or Switzerland has (*). That's fine. If y'all want to press for a Constitutional amendment to change our jus soli policy, go for it.
But don't go around citing English translations of Swiss legalists (i.e. de Vattel) and argue that what they had in mind is what the Founders had in mind. And don't forget that Justice Antonin Scalia has publicly expressed his disdain for "international law": "...I do not use foreign law in the interpretation of the United States Constitution."
The US is not the UK. Justice Scalia used to play poker with a late buddy of mine.
Who was Justice Gray? "On 1881, President Chester A. Arthur nominated Gray to a vacancy on the Supreme Court of the United States; he was confirmed the following day[.]"
Who was President Chester A. Arthur? Chester Arthur was a British subject at the time of his birth in Vermont, because his father was had not yet a US Citizen.
Arthur was not an elected President. He became President when a fanatic supporter of Mr. Arthur assassinated the then-President James A. Garfield.
Why would a fanatic supporter of Mr. Arthur kill President Garfield? Because the killer, Charles J. Guiteau, wanted a job under the Spoils system, which he knew--as did all of America at the time--a system of which Mr. Arthur was close to the top. Mr. Arthur was, well, corrupt. A corrupt official.
Arthur was the chief underling of political boss Roscoe Conkling. Arthur had gained the most politically enriching position of his era when he was appointed by President Ulysses S. Grant to be Collector of the Port of New York in 1871.
Arthur held that powerful position until he was booted from it by President Rutherford B. Hayes in 1878, in a wave of political reform. The Republican Party supporter of the Spoils system were known as the "Stalwarts" and they remained very influential and powerful. Consequently the candidate Garfield negotiated with them for support by granting them the VP slot. Arthur took it, and Garfield-Arthur won the election.
Garfield disliked Arthur, and refused to let Arthur into his house.
This man, Arthur, was the man who appointed Judge Gray.
In your mind then the Founders equated being a subject of the king with being a citizen of a republic? The history of the founding is completely to the contrary. Jefferson, for example, in his drafting, specifically rejected the word “subject” and replaced it with “citizen.”