First, I'm not aware of the Khazar hypothesis ever being the consensus in the 19th century. It was first seriously advocated then, but it wasn't ever a really widely accepted historical hypothesis. To get a sense of the sort of thing that advocates it, it was written about by Ernest Renan, who theorized it in Le Judaisme--Comme Race et Comme Religion. Renan isn't really a historian--he was a philosopher. And indeed he really just asserts that European Jews are Khazar converts, without doing anything to back up the idea. There's no meaningful historical argument there.

There are ideologies associated with promotion of the Khazarian hypothesis. Interestingly, they are both antisemitic and anti-antisemitic in origin. The anti-antisemitic ones use the argument that if Jews in Europe, racial antisemitism is moot, since the pseudo-science behind it is rubbish if Jews can be conclusively proven to not be an ethnic group. This logic is explicit in Koestler's book on the subject (he's not a historian, either), and seems to be in Renan's ideology too.

The other is antisemitic. A quick google of "Khazars" will find a wealth of people who see the possiblity that Ashkenazi Jews are actually Khazars as evidence of a Jewish conspiracy. This is often articulated politically, with "the Jews are making up their ethnic origins to take land from the Palestinians". These sorts of arguments are common, but relatively easily identifiable and de-bunkable.

With regard to the stars of David, it's unlikely that is is actually a Jewish star. Use of the symbol as a Jewish identifier, rather than simply a decorative motif or a Kabbalistic symbol, is relatively "recent", it's medieval. Also, the hexagram is an extremely common symbolic motif, found in a variety of cultures. With Khazar ones, they're more likely sun disks. That's because similar disks with 5 or 7 rays exist too, indicating that the number of points (which is the defining characteristic of Jewish stars) wasn't the key design element. They also disappear from Khazar graves after the conversion event took place, indicating that they're associated specifically not with Jewish identity.^1

Your implication that ideology makes historians not consider the hypothesis is deeply problematic. First, it's factually incorrect. Historians have written about it. They just haven't gotten much acceptance. Second, it implies that historians are unwilling to take positions that are ideologically troubling, which isn't borne out my facts. Third, it hints at some sort of suppression among historians, which is a rather fanciful claim.

There is, in fact, plenty of evidence against it. The first is language. Yiddish is a Germanic language^2 not a Turkic one, which is what the Khazars spoke.^3 The term Ashkenazi itself is derived from Central Europe, and is a rather clear indicator that Eastern European Jews saw themselves as being of the same population. And there's no evidence that the Jews of classical Ashkenaz saw things any differently. There's also genetic evidence against it, though one study of several gave the hypothesis support in a limited form^4

But most importantly, there's just not that much evidence for it. Even Wexler's linguistic claims (which I rant about in footnote 2), if accepted, only show that Yiddish is Slavic, and a sort of Slavic that is far geographically from the Khazar's stomping groups. Often people's layman's reading of articles in news outlets about scholarship on this makes it seem like there's a huge body of information for it, but there isn't. Even Wexler himself admits that the hypothesis "suffer[s] from a dearth of convincing evidence". Or in the words of Paul Golden, "Perhaps one of the problems is that there are few statements that can be made about the Khazars without resorting to qualifications. They were and remain problematic."

So historians don't accept it because it's a hypothesis that upends conventional wisdom in a way that requires substantial evidence, evidence which is entirely lacking. The fact that academics have gotten published arguing otherwise, and had their findings widely reported on, shows that the suppression you hint at in your OP is not significant.

1. Brook, David Alan. The Jews of Khazaria, 2006

• There is a somewhat well-known paper arguing otherwise (which is explicitly connected to the Khazar hypothesis), there's very little evidence for it. Reading the paper it is immediately clear that Wexler is looking very hard for evidence in the language, rather than letting the linguistic evidence speak for itself. He constructs entire grammatical paradigms which are totally unattested just to support his hypothesis. Further, Wexler entirely ignores the dialect continuum in Yiddish, which shows quite clearly that the language can exist independently of Slavic loans. There are no attempts to meaningfully interact with the counterpoints of other linguists. Tellingly, his primary source is his own earlier work, which should give insight into how not mainstream this theory is. And, of course, his argument is that it is a Slavic language is only vaguely connected to the Khazarian hypothesis, since the Khazars were turkic-speaking. Another thing that jumped out at me is his unquestioning mention of the "fact" that Modern Hebrew is relexified Yiddish, which is similar--it has a few big proponents, but is not widely accepted, and has scant evidence for it.
• Erdal, Marcel. The Khazar Language. From The World of the Khazars, selected papers from the Jerusalem 1999 International Khazar Colloquium
• This study specifically had the massive oversight of using Armenians as the "Turkic control", which is a somewhat perplexing error, as Armenians are not a Turkic group. This is Elhaik's study. Others, such as Hammer et al, show a chunk of Middle Eastern DNA in the genetics of Ashkenazi Jews. The sources cited in this wikipedia are a decent place to start. So is geneticist David Goldstein's text Jacob's Legacy: A Genetic View of Jewish History, which synthesizes a great deal of research on this subject.