Don’t get too excited. This applies to someone who was explicitly immune to U.S. law. Illegal immigrants have no such immunity.
That Vattel is cited shouldn’t mean much; no-one debates whether he was influential to America’s founding fathers. Vattel never uses the phrase, “natural-born citizen.” There’s little basis for the assertion that the French word, “naturel” means “natural-born citizen” as opposed to simply, “native.”
Vattel’s book on natural born citizenship was a favorite of George Washington. He failed to return it to the library. Nevertheless Vattel was cited by the founders and lawyers in early case law on citizenship
The French word “naturels” used by Vattel translates to “natural born”. See: https://cdrkerchner.wordpress.com/2023/04/15/my-translation-of-a-key-sentence-in-emer-de-vattels-1758-treatise-on-natural-law-in-section-212-des-citoyens-et-naturels/ ... and ... https://lonang.com/library/reference/vattel-law-of-nations/vatt-119/