You had it right in the first part of your post. Most "Jewish" gangsters in the American Russian Mafia are ethnic Russians who used fake Jewish internal (to the USSR) passports to escape under provisions of the Jackson/Vanik Act of 1974.

The prime example is Marat Balagula, who was the top Russian mobster in Brooklyn for a while. He claimed to be Jewish on his exit application. But "Marat" is not a Jewish surname. "Balagula" is slang for "gangster," or "roughneck." The clever bastard put "Marat the Gangster" on his passport application, and the Soviet authorities bought it.