What mechanism was most important in Stalin's rise to power within the Soviet political system?
I would argue that the most important mechanism by which Stalin rose was his ability to gather political power to himself. This was done largely through bureaucratic means (as opposed to via charisma or force).
Stalin made great use of his position as general secretary of the party. This had been seen as merely a bureaucratic, clerical post, but Stalin used it to amass power. For example, he had control of who entered the party and who got promoted within it. This allowed him to build a large group of lower-level party members and functionaries who were loyal to him (because he was responsible for placing them in their positions).
By bureaucratic mechanisms such as this, Stalin took power in the Soviet Union. He was later to consolidate and hold power via terror, but his original power came from his bureaucratic and administrative abilities.