Following your krumme logic that Rashi, as an Ashkenazi, must’ve come from the Khazars, you retorted (direct quote of you:) “I wouldn’t respect Rashi less if he were a descendant of converts.” Thus implying that the Khazar converts were kosher geirim, not your new tune of today that they (and implicitly Rashi and the Baalei Tosfos too) are a “scourge”.
P.S. The Rabad, Rav Avraham ibn Daud (a Sephardic godol), mentions encountering rabbinical students descended from Khazars in Toledo, Spain in the 1160s. Here’s a Sephardic source that clearly the Khazars merged into the Sephardic world. Hence you have no and can produce no evidence that they melted into the Ashkenazic world; only strong evidence that the Khazars melted into the Sephardic world.