GenZedong

The two chiefs issues are the pre-genetics claims and the genetics claims.

The pre-genetics claims were hand-wavy guesswork that antisemites latched onto rapidly and then some anti-Zionists reflexively used because they wanted to undermine Zionism (using a bad argument, as I argued). Israel's conflation of Judaism and Zionism has often created situations in which there are varying degrees of antisemitism used against Zionism, ranging from explicit and raging antisemitism to casual tropes to simply mixing up Judaism and Israel when making criticisms. Several anti-Zionist groups, including some Soviet ones, latched on to the poor pre-genetics evidence and ran with it for political reasons, for example.

The genetics research is fraught. Comparative genetics is complex to analyze and very sensitive to the method used and assumptions made. There are scientists who claim that Ashkenazi Jewish population data suggests origins roughly in the area of Turkey to Palestine and this is generally the most popular interpretation. It certainly has decent evidence. At the same time, there are others who do see ambiguity there and markers that suggest ancestry near the caucuses as well, and perhaps unsurprisingly, Slavic. Ashkenazi Jews are certainly the result of diaspora, the only mystery is exactly where it started, so it's challenging to tell the difference between "the diaspora started here" vs. "the diaspora moved here for a while and then continued". From my perspective (and I do know a decent amount about the general methodologies), it seems like there are not enough seminal studies on the topic to properly challenge either hypothesis and it's also difficult to disentangle from scientists' biases, as the Khazar origins hypothesis has this history with antisemites and most people are unwilling to touch on it with ambiguous data. Some of the scientists who did, though, were Israeli, for what it's worth.