Specifically, uncertainty in potential results. Swinginess. Random happenings because the dice get a mind of their own. That sort of thing.
Agreed. The finale of the pilot of Star Trek: Deep Space Nine and it's discussion of what makes baseball entertaining to humans is very relevant here.
I have played and like some "story" games, but one thing many of them lack is uncertainty. Their mechanics tend to favor participants being able to say things that become true in the fiction (even if they don't call it that).
I think that "story" and "indy" games are too broad of a category to lump them all together like that. There are some that aren't swingy enough and there are some that are too swingy.
What to me is more interesting is the actual metagame that is created around the players trying to overcome the swinginess and achieve the desired result of their play. That is to say, given that the game involves a certain amount of luck, and given that we assume that while the player accepts the possibility of failure, they don't want to fail and are so motivated to minimize the chance of failure, what gameplay does the game provide for that engages with that play (whatever it's aesthetic motivation, whether self-expression, desire for a satisfying narrative climax, or challenge).
Trad games to me provide for at least the possibility of social, functional metaplay around trying to achieve success. For example, Gygaxian gaming involves "skilled play" of a tactical variety where in game choices offer a chance to mitigate against the luck of the die.
But to me most story games don't provide for social functional metagame around trying to achieve success. This may or may not be deliberate, but the lack of provision for that is often dysfunctional in practice and often creates a game which plays very differently than the designer's apparent intention. For example, in trad gaming we can be aware of the DM playing favorites and adjudicating two different player's propositions differently, favoring one player's propositions for out of game reasons. Or we may be aware of a player browbeating and wheedling a DM into giving them what they want, up to the level of bullying a DM into giving them the resolution that they demand simply to keep the game going. We typically see these as examples of anti-social and dysfunctional play.
But in practice with story games, the mechanics are often basically inviting these processes of play to be the actual processes of play. The actual odds of something succeeding are almost entirely dependent on convincing the GM that they should happen and only by convincing the GM that it should happen can you improve your chances of success and mitigate randomness. While that is true to some extent of most games (as practical matter, the GM always sets the DC), story games often have such vague guidelines and simplified mechanics that metaphorically batting your eyelashes at the GM is the only way to mitigate against randomness.