Carrie Rickey has a fabulous article in the Forward documenting Hollywood's history of casting non-Jews to play Jews (alongside the famous propensity of Jewish actors to change their name in a goyish direction -- paging Jonathan Stuart Leibowitz). I was a bit surprised it didn't mention the most recent example of this -- the non-Jewish Rachel Brosnahan cast as the lead in the exceptionally Jewish Marvelous Mrs. Maisel. I did recall reading at least a few murmurs of discontent about this, though the general vibe among the Jews I know is that it's a great performance on a great show.
Anyway, the paradox Rickey identifies in her article is that many of the Hollywood moguls who resolutely refused to cast Jews in Jewish parts (and often tried to avoid Jewish themes altogether) were themselves Jewish. What gives? The answer is that these Jews were convinced "that movies about Jews would incite anti-Semitism" -- they wanted at all costs to avoid the sense that they were tribalistic, or insular, or that Hollywood was (as it was in the antisemitic imagination) a "Jewish" front. One upshot of this was that the people freest to produce movies about Jews were the non-Jews. Another, of course, was that non-Jews were considered less objectionable or dangerous choices to play what Jewish roles their were.
There's actually a parallel to the academic world here. Academia in the United States has certainly had a robust Jewish presence, but for many years these Jews almost never wrote on Jewish topics (see here for how this played out in Anthropology). Jews were in fact specifically counseled to avoid such matters, lest they be seen as provincial or tribal. And so for the most part, we wrote on other things. Even now, when there's been a flowering of "identity" research in academia (e.g., "ethnic studies"), Jewish Studies have lagged considerably behind -- again, a fact belied by the raw numbers of Jews in academia (on that score, we continue to be just fine). I can tell you I've been counseled on more than one occasion -- albeit with varying degrees of explicitness -- to downplay or cover the Jewish elements of my research agenda. So this is ongoing.