I would recommend to you A Glimpse Into History: How the Treaty of Kars was Signed by Dr. Ulchenk a look into the diplomatic side of the negotiations https://dergipark.org.tr/en/pub/ras/issue/47562/600744

But basically we can look into a number of reasons why attempting to pursue Kars by military means would have been unwise. Firstly the Soviets didn't have a strong grasp on Armenia itself initially, from 1918-1920 as the Republic of Armenia and the Republic of mountainous Armenia resisted the Red Armies efforts to establish Armenia as a Soviet Republic with the Mountainous Armenians continuing to resist into mid 1921. Likewise to the east in rural regions of Azerbaijan in particular there was continued resistance there to the Soviet expansion into the region while further north Georgia, the Kuban People's Republic and Mountainous Republic of the Northern Caucasus also were all greater priorities for the Soviets than elsewhere. So that's a major reason why the Soviets weren't pressing their extended line weren't pressing any further south in those initial years.

In 1921 of the Soviets looking to bring their various conflicts to an end. Some of their smaller cases its a matter of mopping up the last areas of resistance where as in other cases its mostly a matter of cutting their losses in some places and bringing the conflict to an end. And so the treaty of Moscow and the Treat of Kars is made in order to firmly establish a workable border. Just as in Estonia or Poland at this particular time the Soviets were generally willing to give up some imperial territory in order to establish peaceful relations.

Then as we move into the Stalinist era in the early 20s with Lenin's death we more firmly move away from the exporting the revolution ideological and so Soviet forign policy and military expansions and attempts to do so become more motivated by achieving a strong strategic position as opposed to spreading communism globally.

During the mid 20s trade was becoming an increasing consideration. In order to fund the steady industrialisation efforts of the Soviet Union grain exports were being used, these fell off a little 1927-9 before ramping up a lot again in 1930 but access to the black sea was an important part of ensuring good access to the imports and exports that the Soviet economy were seeking in order to grow its industrial strength. And of course war with Turkey means losing access to the dardanelles and bosphorus.

Another consideration is even with Transcaucasian Socialist Federative Soviet Republic firmly under control was the boarder with Georgia and Armenia it perhaps wasn't as easy to connect to logistically compared to doing so along the European boarders of the Soviet Union along the Northern European Plain.

Coming into the next wave of major expansions in the 1939 and early 40s we see that a lot of these were motivated by securing as strong a position in eastern Europe as possible. Poland's invasion was to a certain extent Stalin taking advantage of the situation with Germany to settle the personal score he had with the Polish as well as adding extra territory to their western frontier to give them additional breathing room. The Baltic state's likewise both provided territorial breathing room between the Soviet heartland and other threats and allowed the soviets to seize control of that industrial potential. Finland while the Soviets did seek to establish a Soviet puppet government if possible initially went into the war and in the end came out of it with territorial expansions specifically pertaining to the control of land surrounding Leningrad and the control of Islands in the Gulf of Finland.

As the Soviets did not really expect there to be an conflict in the immediate future coming from the direction of their southern flank Kars did not have the same kind of strategic value that might have made Stalin want to pursue it during this period.

Later on in the post war period the Soviets did consider attempting to revoke the treaty of Kars and invade the region or even going further to establish a larger buffer zone for its defence in the south. This also coincided with another source of tensions surrounding Turkey baring Soviet Naval Vessels passing through the straits between the black sea and the med. The Soviets had also become dissatisfied with the fact that Turkey had allowed Axis auxiliary vessels to pass through the straights during world war two. Anglo-American pressure against the Soviet Union kept these plans in check though for the next seven years. In 1952 with Turkey becoming a full NATO member the cost of conflict with Turkey would have been almost certainly higher while by the next year both Stalin died making such an aggressive invasion less likely, the Soviet government then chose to back down from its claims and tensions largely died down.