I started responding to this piece by Emma Green on Facebook and before I knew I had written several hundred words, at which point I decided this would be better as a blog post.
I feel like this article asks the wrong question, and the reason it does so is that in American terms, "whiteness" has become a proxy for "legitimate member of the body politic." This formulation makes sense in some ways because the cleavage between white and non-white has been the enduring fault line running through American history. But it's also short-sighted because until very recently (in historical terms) people whose skin color wasn't really a contestable issue as such (e.g. the Irish) didn't really "count" as "white"; their integration into the mainstream of American society was accompanied by a sort of retroactive granting of whiteness while European racial divisions remained as fraught as ever.
Jews never properly fit into this dynamic; the author gropes at this problem but fails to apprehend it because, again, the framework she's working from is the American framework of "whiteness" and "color." But the knock on Jews was never that they weren't "white" or even, I think, particularly that they were somehow racially flawed as such; that was Hitler's particular innovation but again, there's a lot of recency bias here making us think that it was always this way. But if you take a much broader historical view, one of the things that comes up, for example, is Christians using "Jew" as a broad category for "alien," even in contexts where there were not even any actual Jews around to reference. The racist pseudo-science of the 19th century allowed for the possibility of grafting these views onto what was then perceived as "biological reality" but I think this was a convenient fusion of disparate streams of thought.
A much better framework for understanding this would be to read something like "Two Hundred Years Together," Solzhenitsyn's deeply anti-Semitic and yet also non-fascist history of Russian-Jewish relations. It's not that Solzhenitsyn wants to send the Jews to the death camps or thinks they're vermin, it's that he fundamentally doesn't believe that they can truly be at home within a Christian society. There's more than a little parasite/host metaphor going on there, but the crucial point for me is that it operates in the long-standing tradition of viewing "Jewish" as synonymous with "fundamentally Other." What could be more Other than the Wandering Jew, cursed with immortality for taunting Jesus and unable to find a home in any country of the world? The Jew's fundamental rootlessness means that he can never truly be a citizen of anywhere, which is why it makes sense that many Jewish socialists and communists were purged by the USSR under the guise of being "rootless cosmopolitans," who can't possibly be trusted to be good citizens.
Viewed through this historical and trans-national lens, I think the question becomes not "are Jews white?" but rather "can Jews fundamentally belong to any place at all?" Or, for me as a Jew, the question is "can we ever have something other than provisional acceptance as Jews?" Because as we're seeing, what can be made white can be made un-white (unmade white?) as long as the conception of the Jew as alien continues to underlie mainstream society's attitude towards Jews.