Actually, New York accents have been tolerated in the movies for a long time, even if Jewish identity was often hidden. The classic case was bronx-born Jew Bernard Schwartz, aka Tony Curtis, who is forever ridiculed for "yondah lies da castle of my fadda." He got his start in the 1940s.

Since Theda Barra mostly acted in silent movies, presumably her accent did not matter. Her married name would have done away with the obvious Jewishness of her original name, but I guess Bara sounded more mysterious than Braben, and she had already adopted Bara by the time she married. She did make one film using her original name, according to the IMDB.