Is it permissible to question C.I. Scofield's notes? 😄
Also keep in mind that the Ten Tribes were scattered all over the Middle East, so many Israelis could well have descended from them. A lot of European Jews would have descended from their forbears in Israel (tribes of Judah, Benjamin, and Levi). When Jews went into the Diaspora after 70 AD, some would have traveled to Europe eventually and settled there (where they encountered persecution).
Even if they are not Jews ethnically (such as the Khazarians), once they embrace Judaism, they are regarded as fully Jews by other Jews.