That there are genetic whosawhatsits is true, but not very relevant. A convert is Jewish, whether he’s white, black, or purple with polka dots, because a rabbinical court accepts him as Jewish after testing his sincerity and knowledge, and after initially discouraging him, and because he has gone through the ceremony of conversion, including circumcision, and is committed to practicing the religion of Judaism as that is delineated in the Torah.
The Khazars were not genetic Jews. Neither was professor Abraham Kotsuchi, the first Japanese convert. Neither was Rabbi Akiva, the son of a convert and one of the great teachers and saintly figures in the Talmud. But they were Jews all the same. Sooner or later, their descendants mix with the Jewish gene pool and become genetic Jews as well.
You can LARP as one, but you’ll never be Jewish. They still won’t let you immigrate to Israel.