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Are you suggesting that Rabbi Yehuda HaLevi made up the whole story? Are you suggesting that the correspendence between Khagan Joseph and Hasdai ibn Shaprut was fake? Are you saying that they Schechter Letter is a fake? Are you saying that the Kievan Chronicle randomly called the Khazars "The Jewish Kingdom?"

Nobody disputes any of this. What is in dispute was A. whether conversion to Judaism spread beyond the royals and nobles of Khazaria to the general population and B. regardless of A, what fraction of modern day Jews are actually descended from Khazars.

There isn't much evidence of a Central Asian "genetic signature" among Jews. The male-inherited Y-chromosomes point of Middle Eastern origins (similar to Druze, Lebanese Arabs), the female-inherited mtDNA points to European origins (related to Italians). If most modern Jews were Khazars, you'd expect their haplotypes to cluster with Georgians, Armenians, etc, which they generally don't.
Actually, a study came out last year saying exactly this. However, Dr. Eran Elhaik, didn't seem to know when Khazaria existed, where it existed, or what populations it included. Hence the study, The Missing Link of Jewish European Ancestry: Contrasting the Rhineland and the Khazarian Hypotheses. The study would be so bad as to be a parody of misuse of genetics, were it not publicized by low IQ and lazy journalists, Nazis, leftist antisemites, and Muslims. The study basically showed that many Ashkenazi Jews actually do have genetic similarities with Georgians and Armenians. Given that Armenia was called "Ashkenaz" in the Talmud, this is almost a self-evident point. But neither Georgia nor Armenia were ever part of Khazaria. This does not mean that Armenian and Georgian Jews did not move to Khazaria to flee Byzantine and Muslim oppression.

I wish I had been paying attention last year to point this out from day 1.