Subjects were held to be in perpetual allegiance to the English monarchy.
Expatriation was and is a right that was given an extraordinary amount of consideration by our Founders.
Subjects could not renounce their allegiance, therefore expatriation was not possible. Once a subject of the King of England, always a subject of the king of England, no matter where.
Our Founders did in fact renounce their allegiance and were met with war because of the dispute over it, not once but twice.
The meaning of natural-born citizen and natural born subject do not equate, not then, certainly not to our Founders who were for the most part considered natural born subjects and were under the threat of imprisonment or death for challenging it, and not now.
The treatise that spells out legalities as far as citizenship under a constitutional republic was by all accounts quite popular with those same Founders. Lo and behold, it contains a definition of natural-born citizenship that has been cited nearly word for word by numerous Supreme Court justices and several prominent members of Congress when proposing Amendments dealing with citizenship.
Oddly, this definition is rejected by some, in favor of a term pertaining to a feudal relationship to a sovereign monarch, the very same monarch explicitly rejected by those Founders.
Strange.
And which treatise might that be? Certainly couldn't be Law of Nations, since there was no constitutional republic on the planet when Vattel died.
The US created the very first written constitution and therefore the first constitutional republic.
Vattel was writing for a European audience. The British government of the time, for all its flaws, was more free than any European government with the arguable exception of some of the Swiss cantons and the Netherlands. Or perhaps Poland, if you were an aristocrat. Of course, the Polish aristocrats were at the time in the process of destroying their nation with that freedom.