Modified from an earlier answer

Although predominantly Russian in origin, the leaders of the Soviet Union did not really conceive of the Soviet Union as a continuation of the Russian empire. Instead, they saw Soviet identity as supranational in nature. The diverse ethnic composition of the Soviet Union compelled the Soviet state to deal with the national issue and try and defang it lest it interrupt the Soviet experiment.

The overarching rationale behind Soviet nationality policies was a mixture of ideological principles and short-term expedients. The Bolsheviks had outlined their principles in 1913's Marxism and the National Question written by Stalin. The previous year, members of the Socialist Revolutionary Party had endorsed an Austromarxist solution to the the nationalities issue. Formulated by Otto Bauer and Karl Renner, Austromarxism upholds the principle of national personal autonomy, a rather nebulous term that basically means that an individual has a right to declare their own nationality and would join in a federation of their respective national Marxist party. In August 1912, Austromarxism had received a significant boost when the Menshevik wing of the Social Democrats (SD) meet with the Bundists (a Jewish socialist labor group) and endorsed an Austromarxist solution for the Russian empire after the eventual revolution. Lenin, as a leader of the Bolshevik SD faction, saw it was essential to combat Austromarxism and differentiate the Bolsheviks from the larger movement. Lenin tasked Stalin to come up with an official position on the nationalities issue; one that would be congruent with Marx, serve the needs of revolution, and condemn the Bolsheviks' ideological rivals. Stalin's Georgian background added a non-Russian luster to this position paper. Lenin wrote to Gorky "we have a marvelous Georgian who is busy working on a long article 'illuminating' the proletarian solution to the national problem."

The resulting paper was the famous Marxism and the National Question that attacked the principles of national personal autonomy and Austromarxism as a deviation away from Marx. The fundamental difference between the Bauer/Renner paradigm and Stalin's was that Stalin strongly asserted that a nation needs a territorial basis and must reside within a given territory and have a historical claim to that area. Austromarxism was thus a retrograde step according to Stalin as it diffused the strength of a Marxist political movement and would hamper the ultimate goal of Marxism after a revolution: the creation of a new socialist consciousness based upon class, not bourgeois concepts like nation.

The rub for all this is that the Bolsheviks seldom had any clear and coherent policy with regards to the national question, much of it was ad hoc and conditional. Stalin would at various points between 1913 and 1917 assert a quasi-Austromarxist solution and Lenin was really ambivalent about Marxism and the National Question's paeans to national culture on the basis such sentiments were too bourgeois.

This ambivalence imparted to the Bolsheviks a degree of ideological flexibility and that made up for their lack of coherency during the Revolution and Civil War. Lenin learned with the case of Finland that sanctioning national independence too soon could be counterproductive to Bolshevik control. The Finns had appealed to dissolve its personal union with the Russian state after October Revolution and Lenin thought that by granting the Finns this privilege, they would form a common front with the Bolsheviks. This did not come to pass and the Finns tacked to an independent course. Lenin became determined not to repeat his mistakes with Finland when he approached other national movements. Thus the Bolsheviks clamped down harshly on the Ukrainian Rada's attempt to form an independent Ukraine in the name of revolutionary unity. Yet at the same time, the calls for national independence were a powerful tool in the Bolshevik's appeal in the Civil War. While they were repressing Ukrainian independence they were simultaneously proclaiming to the peoples of Central Asia that the Bolsheviks were ending the tsarist "prison of the peoples." The creation of the SSRs was a way to co-opt and mediate the emerging nationalism that the Revolution had engendered.

In its public propaganda, Moscow portrayed the new state structure as the antithesis of the tsarist state, The early Bolsheviks enacted a policy of kornenizatsiia (indiginization) in which the central government actively sought to promote native cadres in these new territories. Lenin and the other Bolshevik leaders were ambivalent about Russification and the promotion of Russians at the expense of other peoples. There was a fear that if the new USSR could fracture if it repeated the mistakes of tsarist Great Russian chauvinism. To counter this, the state promoted the local languages, cultural programs, and most important for Moscow, the strengthening of the local Communist Party establishments. However, Moscow's definition of both "nation" and its "natural" territory were often quite arbitrary and seldom reflected the complexities of national identity in such a vast and heterogeneous polity like the Soviet Union. For example, the Soviet regime tended to perceive ethnic Poles who refused to assimilate to the new ethnic republics (largely Ukraine and Belarus) as either potential agents or dupes of the Polish anticommunist state. The flexibility with regards to the geographic boundaries of Poland helps explains Stalin's Polish policies during the Second World War. He would justify the western shift of Poland as just correcting Poland to its natural historic boundaries and population transfers (aka ethnic cleansing) as an extreme, but logical, application of Marxism and the National Question.

The latter half of Stalin's rule saw a shift of Moscow towards an acceptance of Great Russian cultural hegemony and a functional Russification of the vast Soviet territories. The Second World War reinforced these tendencies as the Soviet state increasingly instrumentalized the weight of Russian history and culture in its war against the fascist invaders. Great Russians became the elder brother in the Soviet family of nations, a subtle, but important shift signalling a Russocentrist orientation of the state. Yet, the nationality principle remained intact despite this and moves towards national distinctiveness ebbed and flowed throughout the rest of the Soviet period. The SSRs became something of a landmine embedded within the structure of the Soviet Union; by creating a structure so integral to the conceptualization of the Soviet Union, Moscow could never really eliminate it.

EDIT: One other thing to add is that Kaliningrad exclave, which the OP references in the question, was very much an atypical example of the Soviet rearranging of national territory. Stalin saw the Soviet acquisition of Königsberg as a strategic priority for the Soviet Union as he thought the USSR needed an ice-free winter port on the Baltic. In meetings with the Allies, Stalin would claim that this portion of East Prussia was "ancient Slavic soil," and this territory would be Soviet. Although the Western Allies initially agreed to this annexation, the Königsberg issue rose up again at Potsdam when Truman and Churchill floated the idea of an Allied Occupation of East Prussia that would oversee a German administration. Stalin outright refused and boldly asserted that the Soviets would through out any provisional German administration and this Slavic soil belonged to the USSR by virtue of the blood shed liberating it. While Poland and the Lithuanian SSR received the southern portions of East Prussia, Königsberg became the focal point of a separate oblast. Rather than having it incorporated directly into the Lithuanian SSR, Kaliningrad became a part of the RSFSR. Again, security reasons predominated this as Kaliningrad became the home base of the Soviet Baltic Fleet and Stalin was leery of allowing the newly created Lithuanian SSR access to these military installations. The real collateral damage from the Soviet's preoccupation with security was the oblast's remaining German population, who became seen as a potential fifth column in this military zone. The result was the USSR expelled these Germans in 1947 and brought in Soviet settlers, many of whom were ethnically Russian.

Sources

Carrère d'Encausse, Hélène. The Great Challenge: Nationalities and the Bolshevik State, 1917-1930. New York: Holmes & Meier, 1992.

Martin, Terry.The Affirmative Action Empire: Nations and Nationalism in the Soviet Union, 1923-1939. Ithaca: Cornell University Press, 2001.

Smith, Jeremy. The Bolsheviks and the National Question, 1917-23. New York: St. Martin's Press, 1999.

_. Red Nations: The Nationalities Experience in and after the USSR. Cambridge: Cambridge University Press, 2013.

In an essay title Marxism and The Nation Question, Stalin spoke out against nationalism calling it, "crude chauvinism," and called on the Social Democrats (what we would today call Democratic Socialists) to, "work solidly and indefatigably against the fog of nationalism, no matter from what quarter it proceeds."

To Marxists, nationalism in the face of imperialism is generally seen as a positive (see North Vietnam 1946-1975). However, nationalism in the sense that the national should be homogeneous is seen as a tool to divide the working class and impede the worker's movement. Thus the reason why the USSR did not claim to be an ethnicity russian state.

Stalin, J.V. . "Marxism and the National Question". https://www.marxists.org/reference/archive/stalin/works/1913/03a.htm

Russian revolutionary government itself was mostly of non-Russian nationalities.

Lenin was born to Mordvin father and Kalmyk mother.

Stalin was Gerogian.

Beria - Georgian

Trotsky - Jewish father, Ukranian mother.

Dzerzhinky - Polish

These are key figures - non of them ethnically Russian.

There were secondary figures, like Budennyi, Molotov, Kalinin, but still...

Also - unlike in Germany, Russia was very multy-ethnic to begin with, and everyone was pretty mixed up, so to start Russian-only state, you'd have to define - who is pure Russian? I bet, there would be very difficult technicalities to define this.

Sources of ethnics of Russian revolutionaries are all in Wikipedia.