During the Soviet era, all religious bodies had to be registered and approved by the state, so the state could also control them. Some Baptists were suspect as tools of American capitalism because John D. Rockefeller, a Baptist, funded a lot of foreign missions. Religious bodies that, on faith and theological principles, refused to register, operated underground. I didn’t realize that that policy survived the Soviets.
I am Orthodox, but I have to agree that what a lot of people say about the Russian Orthodox Church— or at least its leadership — being a tool of the Soviets —has a great deal of truth. St. Tikhon, the last Patriarch of Moscow before the Soviets, sent a message to the Russian Church in diaspora, informing them of the dire situation for the church in Russia and warning them not to believe anything else that came from the Moscow Patriarchate. That led to a defacto self-governing church in America, and the formation of the anti-communist Russian Orthodox Church Outside of Russia. There were some parishes in America that continued to be loyal to the Moscow Patriarchate, that were known to be communist fronts.
Stalin lifted his boot a bit off the ROC and other religious bodies during WW2 so he could get everyone on board for Mother Russia.