I'm afraid I'm not well-versed in historical scholarship and what is happening (or not happening) in that world.I definitely agree with you that there has to me more exploration of these topics, particularly if Jews wish to get to the heart of their ancestry - ALL of their ancestry, which is clearly more complex than simply "Judean."I'm pleased that researchers have found such a strong element of Judean ancestry among the Y-chromosome results, but I'm not pleased at all about how that finding has lead to an apparent lack of interest and examination of the other results, which obviously have strong place within Jewish ancestry as well.
If the "other" DNA (Q, R1a, R1b, I and MtDNA) was not brought in to the Jewish community by the Khazars, then it was brought in by some other kind of European gene flow.This gene flow seems to be particularly strong among the Ashkenazi Levites.
Frankly, there is such a fine line between Eastern European DNA and Khazarian DNA, I'm not sure we can really tell these apart.Because Eastern Europeans, to a large extent, are descendants of early steppe nomadic cultures which lead ultimately to groups like the Scythians and Sarmatians, and it is from these groups (who were significantly though probably not exclusively R1a) that many Eastern Europeans claim descent.The Khazarian empire arose in the very same region as the previous Scythian empire - the two peoples were obviously connected, though the Khazars may have brought into the area some additional "Hunnish" or North Asian elements (perhaps haplogroup Q)?Unfortunately, until there is more research about the distribution of haplogroup Q in Eastern Europe and Central Asia,I'm not sure whether we can really say it is Khazarian, Hunnish or simply Eastern European in origin.
What I am saying is it is sort of irrelevant whether the Jews obtained their non-Judean DNA directly from the Khazars or from Eastern European Slavs, since the two groups are nearly interchangeable and originated from the same early steppe nomadic cultures.But Khazarian history is a fascinating and important topic for Jewish historians, archaeologists and DNA researchers, and clearly should be explored to it's fullest limits possible.