First, Koestler's work was funded by British intelligence. Second, the book is called The Thirteenth Tribe: The Khazar Empire and Its Heritage. Third, it's not about all Jews, it's about Ashkenazi Jews, which is not the question here. Fourth, he posited that removing a bio link to Ashkenazim would reduce anti-Semitism, so his work has been hijacked. Fifth, it doesn't answer the "Judahites aren't Jews" formula because "Ju" was used in 800s France (obviously of the Judahites) before the Khazars dissipated ca. 1016 (capture of Georgius). Sixth, "The Khazar hypothesis [is] that Ashkenazi Jews were primarily, or to a large extent, descended from converts to Judaism among the Khazars". Primarily or largely, not entirely. "Xue and others note a wholly Khazar/Turkish/Middle Eastern origin is out of the question, given the complexity of Ashkenazi admixtures." The background, and removing the "totality" aspect, are all before the merits.

"The hypothesis also postulates that after collapse of the Khazar empire, the Khazars fled to Eastern Europe and made up a large part of the Jews there", i.e. they integrated just as I said, they didn't stay separate as Khazars. The first attempt to connect Khazars in was for Johann Ewers 1808 to guess some became Varangians in the 800s and founded Kievan Rus in the late 900s (this connection seems to remain the base ever since). By 1883 Ernest Renan exaggerates this into "in all probability nothing or almost nothing that is anthropologically Jewish", still speaking relatively cautiously of course. Isidore Loeb 1885 picks up that argument with anti-miscegenation rhetoric (as if "look, we Jews are mixed-breed, you should be too"). "Jewish settlement in eastern Europe took place very early, and these rooted eastern communities readily accepted into their midst Khazars who had converted, absorbing many thousands into the Kievan empire." Karaites (not Ashkenazim) including Seraja Szapszal also wanted the fully-Khazar hypothesis applied to themselves, again to prevent anti-Semitism (and it worked among Nazis), even though "early Karaite literature speaks of Khazars as mamzerim".

Genetic and other scientific study is about what one would expect for a nonextant populace: namely, yielding numbers between 0% and 100% correlation. Therefore, a reasonable conclusion would be that Ashkenazi have some Khazar DNA and some not, and the amount is disputable. Because of the evidence that in any historic thesis the Khazars interbred with Judahites it would be rational to describe Ashkenazi as mongrelized Jews. That is, my statement they are ethnically Jews is accurate just like black Jews (American, Ethiopian, Lemba) are ethnically Jewish but mixed. Ethnically Jewish does not mean unmixed, it means pure enough in pedigree and practice to be accepted by other Jews.

Now this is Conspiracies so let's try the theoretical version. What if the Khazars were led by a satanist cabal that converted to Judaism to corrupt it? What if the cabal intended to exploit lack of records to argue the chameleon position, ethnically Jewish or non-Jewish as convenient? What if satanism were then a driving force in Ashkenazi and thereby in pressure upon Sephardi and other Jewry in time? There is no hard evidence for any of that, but we could propose it to try to understand the minds of the foaming-mouth anti-Semites. The difficulty would be that even granting that theory it still means the continuing Jewish polity still has the rights to the title "Jewish" even if its control has changed (much like Rome still has the title "Catholic"). Satanic evidence is slim, and a satanist living and dying doing nothing satanic is hardly advancing satanism; so it's more fitting to say Jews and Gentiles have occasional successful infiltrations of satanism that are historically notable. Jews still have the pre-Khazar Talmud that they cannot change one bit of, and they have the Sephardi continuous interpretation stream to contend with that has not been coopted genetically in any theory. Therefore the Christian evangelist can still appeal to individual Jews to recover their pre-Khazar heritage that cannot be hidden by any proposed Khazar mafia no matter how endemic. That heritage is, as the Christians recognized, totally prepared to receive the Jewish Messiah unless he is personally resisted. Therefore, in the most generous reading of evidence, the Jews are still sufficiently ethnically Jewish and they still have an uncorrupted pre-Khazar heritage that they can be restored to (in which case they can be evangelized to Jesus as well). So the great disputation is, as I've said, not about replacement but about influence, and influence can be undone, and that is best achieved without stumbling blocks.

One theory I would propose on the data is that when Jews are early proposing themselves as mongrels it is a natural consequence for the charge to return upon them in spades, i.e. some anti-Semitism echoes Jewish self-deprecation. Logically, whether a Jew wants to self-define as mongrel or pure, neither is an excuse for other races to self-define as mongrel, and neither is an excuse to go beyond demographics to racial hatred or collective judgment of innocent with guilty. Rather, all nations having equal rights of self-determination is a position that rises above all petty disputes.

To the rest, no evidence has been adduced that the Rabbinical Jews formally cut themselves off. There is evidence they cut off certain rabbinical partisans (including continuing proponents of Sadduceeism and some Christian heretics like tritheists), but they used plural formulas to describe God so their doctrine of plural attributes in unity never rejected proto-Trinitarianism.

And, if you know you're quoting it wrong, don't keep doing it. The theological system has none of that, and the Talmudic culture has none of that applied to Jesus but only to the stock character Yeshu who combines 3 or 4 different people of different eras. The Gittin passage has no "currently residing", no "in a cauldron", no donkey, no semen, and a view of hell that is different enough to reject that word too. The Jewish religion does not exist entirely "as opposition to Christianity", one view is that Christianity is just a Noachite pre-naturalization movement on Judaism's behalf. You know my open challenge to anyone to prove that view endemic to formal Jewish religion.

How would you like to work this issue out rather than just repeat easily dismissed tropes that you admit are incorrect?