A minor comment about the beginning of that article. The author introduces the territory of Khazaria and the eventual self identifying Ashkenazis from which the Rothschilds came, but immediately quotes Revelation 2:9. The letters to the seven churches were obviously delivered many centuries prior to the Khazarians' nefarious rise to power, and so does not direclty apply. The letters to the churches addressed specific situations and people groups the original audiences would have been familiar with.

In the case of the Jews-by-name-only reference in Rev 2:9, it's clear from history and other NT passages that these were the so-called Judaizers within the Gentile region churches, who insisted that non-Jews adopt the Mosaic law and circumcision in order to be justified. In other words, they were misusing the whole purpose of the Torah just like the Pharisees before them, and were therefore ignoring the Law of GOd even as they appeared to obey it meticulously. And also in other words, just like the Pharisees they were of a false religion known as legalism which is the "moral" road to hell, a road maintained by Satan since it seeks to mute God's grace in the gospel for sinners.

So they were not true Jews, because they rejected the faith of Abraham their geneological forefather, since he was justified apart from works of the Law (Romans 4, Galatians 3)

All that to say, that Revelation verse has nothing to do with ethnic non-Jews claiming ethnic Jewishness. Different matter.