Exactly. Blackstone was a reporter of British law (and at times British specifics involving international and natural law). Vattel, however, was a reporter specifically of international (and thus, natural) law.
But the question of who a nation's citizens are isn't a matter of international law. Obviously there is an obligation by states to a body of citizens. But the establishment of different grades of citizenship or the exclusion of some native-born persons from full citizenship isn't something natural or international law commands. It's something different society's have to decide on their own.
I could see arguing that a state has a natural law obligation to extend full rights those born on its soil, but not that a state has a natural law obligation to exclude those whose parents weren't born in the country from enjoying the full rights of citizens. So just what a citizen or a "natural born citizen" is, is something countries work out on their own. Not something set in stone by 18th century French legal theorists.
Look at the difference between the Old World, set in its ways, and the New World where the parents or grandparents of so many people came from somewhere else. Vattel may shed some light on one aspect of where the phrase "natural born citizen" comes from and what it means, but one can't turn him into some kind of superlegislator whose view must be accepted if everything else in our system goes against it.