Communism

Yeah, so what you're missing is the liberalization period. The SU under Kruschev and his faction decided to steer the country towards liberal reforms which allowed for private accumulation of capital. It also adopted a strategy of trying to become friendly with the West in a failed attempt to co-dominate the world with them.

In essence, the SU didn't go quickly, it took decades.

Just because they shouldn't doesn't mean they didn't. The Law on Cooperatives, which came about in 1988, allowed for private ownership of businesses in the services, manufacturing, and foreign-trade sectors. The reforms created by Perestrokia set the conditions that allowed for oligarchs to come into power. Take Boris Berezovsky, for example. From Wikipedia:

In 1989, Berezovsky took advantage of the opportunities presented by perestroika to found LogoVAZ (Russian: ЛогоВАЗ) with Badri Patarkatsishvili and senior managers from Russian automobile manufacturer AvtoVAZ. LogoVAZ developed software for AvtoVAZ, sold Soviet-made cars and serviced foreign cars.[39] The dealership profited from hyperinflation by taking cars on consignment and paying the producer at a later date when the money lost much of its value.[40]

There obviously is a lot more historical development that led to this point in USSR history, but Perestrokia really set the stage for capital accumulation, it would seem. Early on in these reforms it seems that many of the oligarchs were also members of the Soviet state; again, see Wikipidia.

The majority of oligarchs were promoted (at least initially) by the Soviet apparatchiks, with strong connections to Soviet power-structures and access to the funds of the Communist Party.[4][12][13] Boris Berezovsky himself was Head of the Department of System Design at another Academy of Sciences research centre. His private company was established by the Institute as a joint venture.[14] Mikhail Khodorkovsky started his business importing computers under auspices of the Komsomol-authorised Center for Scientific and Technical Creativity of the Youth in 1986, briefly serving as a deputy secretary of the Komsomol for a district in Moscow in 1987. His move into banking two years later was funded with the support of Komsomol alumni working in Moscow city government. Later, he served in the Russian government as an adviser to the prime minister and a deputy minister of fuel and power while still running his business.[15] Vladimir Vinogradov was the chief economist of Promstroybank, one of the six banks existing in the Soviet Union,[16] previously serving as the secretary of Atommash plant Komsomol organisation.[15]