The Soviet Union was ideologically committed to internationalism based on the idea of united the working class, but nonetheless had do deal with the concept of ethnicity (or nationality) in the Soviet Union right from the beginning. You're asking about the post-Stalin period, but I think this question needs to go back further because the context was established earlier on and had important ripple effects all the way to the disintegration of the Soviet Union.

Early on the question of what to do about Russia's empire weighed on the minds of Soviet leaders, Lenin included. The dilemma was easy to state and difficult to solve. Soviet leaders wanted to “keep” Russia’s former imperial holdings in the Soviet Union – after all it was a movement of international communism. On the other hand, their ideology told them that nationalism was a force that would oppose an imperial power and that they were, if nothing else, the de facto heirs to the Russian empire. The solution, on paper, was actually the deliberate development of nationalities in a process that created what the historian Terry Martin calls in his book of the same title an “Affirmative Action Empire” in the 1920s and 30s. The idea was, put a little roughly, if we let nationalities develop while we are in power, they’ll understand we aren’t interested in dominating them and voluntarily stay in the fold and we’ll therefore bypass the issue of nationalisms tendency to resist empire. This translated to the development of somewhat strong local identities – sometimes where there hadn’t even really been any before because the Soviet government needed, as a matter of policy, for this sort of thing to be “legible.” There were powerful local elites (Communists, of course) that were at the head of the governments of their various SSRs within the Soviet Union.

Nonetheless, ethnic Russians always dominated the highest levels of power, for example the politburo. This is the short answer to your question. Ethnic Russians, and after that slavs in general, dominated the central leadership of the Soviet Union both before and after Stalin. Of course, Stalin himself was not Russian, but Stalin by no means represented the ascendency of non-Russians in a broader sense within the Soviet power structure. Repression under Stalin was well documented and worked to stifle what ethnic/national dissent (among other kinds of dissent) existed during his rule, but it was barely any more out in the open after Stalin’s rule. Of course, there is a difference between ethnic opposition to centralized Soviet rule and the mere inclusion of non-Russian ethnicities inside of that central power structure. They are related insofar as the relationship between Russianness and non-Russianness was a continual source of difficulty for the Soviet government, and the extent to which the various nationalities were not represented inside the power structure contributed to their eventual movements towards autonomy.

It was not so much that it was absolutely impossible or even something as far as illegal for non-Russians to make their mark on the central leadership. Rather it was exactly what you’d expect with a central locus of power that is self-regulating. In the end, the politburo and Central Committee were largely conservative in the sense that they wanted to preserve the existing power structures that they controlled. Khrushchev’s rule might be have been a slight blip on the radar here, as he was more interested in promoting a little more vibrant Communist Party than had existed under Stalin. This was threatening to a lot of bureaucrats who had a lot to gain from things staying as is. Brezhnev rose, in part, due to a reaction against Khrushchev’s attempt to nudge things in that direction.

It was not until the mid 1980s with Gorbachev that things really began to change. In part, that had to do with Gorbachev being of a different generation and also being somewhat of a “true believer” in the sense that he embraced those very early ideas that I mentioned up front, including about the ostensible autonomy of Soviet nationalities. Brezhnev’s policies had placed the conservation of centralized Soviet power at the top of his priority list. This explains both his policies inside the Party and Soviet government and the so-called Brezhnev doctrine that was invoked time and time again to justify crushing dissent in Eastern Europe. Gorbachev, on the other hand, took an entirely different view, which explains things like perestroika and and glasnost’ that resulted up in the opening up of Soviet society/culture and also in the renewed development of national movements. This, of course, brings us back to the beginning. Ronald Grigor Suny’s book The Revenge of the Past explores precisely this topic, connecting the earliest Soviet nationality policies to the Soviet Union’s demise. The Soviet Union, in this sense didn’t “collapse” insofar as that implies a kind of pile of rubble left over. Instead, the Soviet Union disintegrated or broke down into its component parts, that had developed at least in part into their existing states in by the end of the 1980s due to central Soviet policy from the beginning. When that was combined with the comparative openness of Gorbachev’s rule we saw now the drastic change in the ethnic or national character of centralized power, but the establishment of local autonomous movements and power. This isn’t to say that is why the Soviet Union came to an end, but does a lot work to explain why the aftermath looked the way it did and how that process occurred.