The stories of Stalin's two sons are tragic. Both ended badly. Both suffered from Stalin's bullying and neglect. Vasilii became attached to privileges that his father's subordinates showered on him, and his stubborn insistence that he deserved them caused trouble to the very end. Had he been better behaved and more predictable, the post-Stalin leaderships might have looked the other way with respect to his theft of state property. Deep down, Soviet authorities feared the adverse publicity that an unpredictable son of Stalin could bring down on them.
There is another angle to the story of Stalin's sons to consider: Why did their father not groom them to take over after he was gone? In many cases (Duvalier in Haiti and Aliev in Azerbaijan), dictators extend their reign beyond their natural life spans through their sons. Stalin's handling of his sons makes clear that he had no intention of grooming them as successors. Why was this so? We can only speculate, but it seems clear that Stalin made sure that there would be no consideration of successors during his lifetime. He left it to his successors to fight for his position, just as he had fought to succeed Lenin. Whenever rumors circulated that a particular party leader was a likely successor, Stalin brutally cut them down to size. For those who showed independent intiative and ability, Stalin had them killed.
An able and ambitious son of Stalin would have been considered by Stalin as a threat to his power, and Stalin's primary goal was unlimited and unbridled power. It may even have been true that Stalin, through abuse and indifference, made sure that no son of his would ever be considered as a potential successor.