Thanks for the hints. Maybe I have just not seen a game that ultilized those distribitions well.
If I may, let me give you an example from my Dungeon World game, which uses 2d6.
Sometime roughly in the late middle of last year, the party (Battlemaster, Spellslinger, Fighter, and Bard) wrapped up their preceding adventure to a lost Genie-Rajah city in a volcanic caldera and came back to Al-Rakkah (the main city of the region and their home.) One part of the Battlemaster's ongoing story is that he is investigating the story of his mother, an "elf" who disappeared when he was relatively young, leaving him with his human father. She left him an important sword (long story, won't bore you with details), however--a sword that can grow stronger by absorbing the powers of captured enemy weapons. She was actually one of the "El-Adrin" who hid their civilization away in a pocket plane long ago to avoid its destruction due to a change in the nature of magic.
As a result of the above, the player has been looking for leads on his mother's people, who have been completely forgotten by the modern world. Previously, one of their allies had said he'd heard rumors (which the party acted on for an adventure quite some time ago), and the player said, "Hey, where was Shen going for his information?" We played it out, and it turned out he was talking to an antiquities dealer who has both a legitimate business and some illicit (but not particularly immoral) contacts as well. She's made good money from her business and her marriage to a (very minor) genie noble, maintaining a townhouse in Al-Rakkah, a residence and private gallery in another city, and a private estate outside the city proper.
But the player rolled snake eyes on a Discern Realities (essentially Perception) roll to get information on this woman. (Wisdom isn't one of his strong stats, so Discern Realities rolls can be a challenge for him, but that challenge does not mean the consequences are "bad" in a painful/letdown sense. Just that they're not what he wanted.) That meant he got a "fail," enabling me to make a "hard" move. So I said nobody had seen her in weeks, and there was evidence of foul play--not only would he not get the answers he sought, but something bad had happened and would probably warrant fixing. Their investigation of the woman's townhouse caused them to get jumped by alchemically-roided-out gangers who picked a fight (one of whom actually transformed into a half dragon! Story for another time) and revealed that she had come into possession of something
big, something she knew was dangerous and guaranteed to attract attention. Her disappearing act was partially intentional, but she'd clearly gotten wrapped up in something she wasn't ready to deal with, and the party both wanted to aid her (they're pretty reliably heroic) and find out what all the fuss was about.
From this one, single failed roll, an entire adventure arose, dealing with a time break, fractured shards of possible realities (some of them quite disturbing to the players), mysterious artifact books that can connect to other worlds, and otherworldly beings whose job is to keep the cosmos running smoothly. I planned exactly none of this prior to the player rolling poorly that one time, and a good portion of the adventure arose from what the players chose to do with it (e.g. they didn't try to reason with the modrons, but
did act diplomatically with the Genie-Rajah queen they met in a distant-past time shard) rather than "neat scene I thought the players would enjoy."
All this, from
one roll going poorly instead of going well. That's surprise and variability serving the good of the game to my mind.