That doesn't explain why the founding fathers would use a definition without explanation from a theoretical work casting aside a definition of citizenship familiar to all.
You are assuming it was "theoretical" in their eyes. It wasn't, any more than Blackstone's "Commentaries on the Laws of England", with which it was more or less contemporary. (1765-1769 for Blackstone, 1758 for Vattel). Blackstone's commentaries were presented first in oral form starting in 1753)
There was no such "familiar definition" of citizenship. There were definitions of natural born "subject". But being subjects of some royal ashhole was just what they did not want. Thus they chose, or accepted, a definition that would not allow the first generation progeny of European royalty to hold the office of Commander in Chief of the American Army. Using the "born in the country" definition (that was not really the pure case in England either) would have allowed some "peer" of the Relm to come over with a pregnant Royal wife, or marry some American wench, and then set up, using the ample purses of the "great families" of Europe, his progeny as President and Commander in Chief, and from their form a new Royal Line, in America.
To such a notion the founders, common men and more educated elites alike, would have said not only No, but Hell No.