Right. Practices like the impeachment of Charles I. Thwop! Roll! By Act of Parliament. LOL!
The United States was a new idea. It was not about perpetuating old, inferior European traditions.
The United States was a newer adaptation of some centuries and millenia old "European traditions." Traditions such as the republican form of government founded upon a democratic citizenry exercising sovereignty through delegated authority. The jus sanguinius (right of blood inheritance) and jus soli (right of birthplace) were both ancient principles of law with origins at the very beginnings of civilization so many millenia ago.
The legal principle of natural born citizenship is as ancient as the multi-millenial jus sanguinius (right of blood inheritance) and jus soli (right of birthplace) customs. The Founding Fathers' usage of Vattel's Law of Nations was merely a very timely and useful summary and recapitulation of the already ancient practices as they related to the reaction against the divine right of kings as a source of a government's legal authority versus the consent of the citizens. Vattel translated and consulted earlier works to compile his treatise in comparison to those supporting the divine right of kings.