Russian theater is quite different from American theater and the Broadway ‘scene.’
Instead of a focus on traditional theater, Russian theater was and is very focused on dance (ballet being the most prominent form), opera (with at least a few opera houses in each major city), and people known as ‘national artists,’ who are a major part of the Russian theater community and Russian cultural identity.
Two great examples of this are Vladimir Vysotsky and Yuri Nikulin, an extremely popular singer and actor, respectively, during the time of the Soviet Union. These national artists were household names and were known by, quite literally and without exaggeration, everybody, creating and starring in some of the most popular Russian films and songs. An American equivalent of this would be someone such as Beyonce or George Clooney.
Moreover, the Russian arts community has an intense focus on ballet and opera, creating productions of already existing works (such as the famous Swan Lake ballet sequence or Madame Butterfly, one of the most popular operas at the Russian opera houses). Additionally, the Russian music community was quickly growing (with bands such as LYUBE or Vladimir Vysotsky) and the Russian writer community experienced similar growth with authors such as Sergei Lukyanenko, who authored the World of Watches series (Russian: Вселенная «Дозоров»).
Musical theater, however, was never a focus of the Russian theater tradition. One of the more anecdotal examples of this is the Russian ‘translation’ of Les Miserables, which kept the original English lyrics but translated the spoken words of the characters (in an operetta, which has around a dozen lines of spoken dialogue). Some American musical theater shows were partially translated (most notably the song “All That Jazz” from Chicago and “Cabaret” from Cabaret). To my knowledge, the only musicals to be fully translated into Russian are Man of La Mancha and The Phantom of the Opera.
This, coupled with the Soviet Union’s relatively strict immigration laws, meant that when those of Russian descent did arrive in the United States, they did not contribute much to the theater community, as the community at home was focused either on productions of already existing operas/ballets or on music and writing.