Shalom, and thank you for your response.
I have had this argument many times.
The Wikipedia article does not counter the historical evidence of Jews in the Khazar empire. As to genetic evidence, even that article acknowledges that "there is lack of clear modern descendants of the Khazars that could provide a clear test of the contribution to Ashkenazi Jewish ancestry." For the same reason, the DNA tests of my wife show no Chinese ancestry, even though she has verified it through records and you can see it in her cousins.
The basic problem with that article, and the studies it cites, is that the issue is treated as and either/or Black or White question. We may not know how much of our ancestry can be traced to the Khazars, but it is clear that there were many Jews in Khazaria, that several of the Kagans were Jewish, and that Jews fleeing persecution were welcomed there. To say that there was no mixing over the centuries of Khazarian rule, and that none of those people ended up in the Pale after the destruction of Khazaria by the Rus and their dispersal throughout the Pale, simply defies logic.
The argument seems to be that acknowledging anything other than a direct and pure line between modern Jews and the Patriarchs diminishes the Jewish claim to Israel. I do not agree with the Jewish claim (or anyone's claim) to the land of Israel, but even if I did, I don't think acknowledging anything other than a Levant ancestry should make a difference.
I take pride in the diversity of my ancestry. At the same time we celebrate our heritage, we need to move towards a universality, in which "nation shall not lift up sword against nation; neither shall they learn war anymore.” (Isaiah 2:4).
For more on my views on this, see:
Why Don’t You Go Back To Where You Came From?
https://qcp.medium.com/why-dont-you-go-back-to-where-you-came-from-983098283d3a