Realistically, what game could Twitch play next? Harvest Moon? Fire Emblem?
Direct control of the character
The memes and storylines sprung up from the idea that Red is doing all this *himself*. Look at how often he's represented as a tortured mindless robot in fanart. That creates stories, and stories make this sort of thing thrive.
Inability to completely screw up
People are investing a lot of time in this, and game overs leading to having to navigate the front-end menus has a massive risk of losing *everything*. The fact that death in Pokemon just teleports you to a safe location is a huge boon. I'm not overjoyed about Release being possible, actually; if people weren't scared of it, we'd get more PC usage, which could easily lead to some frankly bizarre group makeups and more willingness to actually catch monsters. If I were to run it again, I'd give serious thought to making releasing impossible. We've probably stabilised now, but I suspect there was a point recently where had Pidgeot been released, that would kill the whole project as people lose morale and give up.
Small random occurrences having amusing results
While it's good that small random occurrences can't result in total failure of the run, having occasional pitfalls is fun. There is a line you have to draw here, though; I've gone on record as saying that the maze was *too much*, while the ledge and trees were just about enough.
...or at least slow enough that lag isn't an issue, but turn-based is probably a safer representation.
Too much linearity means it's just a group of people trying to follow a strict path, and that will get either dull or frustrating to follow. Pokemon has a lot of freedom, particularly after the Rock Tunnel, so random events can have major consequences. Also, of course, it's worth noting that the points that have caused trouble are the points where you suddenly have to play in a linear fashion, and stringing together more than about five-ten differing moves (there's more leeway if 'spamming' fits the model) is beyond the group.
All gameplay challenges can eventually be worn down; it never feels like we've made *no* progress, because even gaining some small amount of experience is something towards future success. When it starts feeling futile is when people will switch off.
My best guess, here - other than other Pokemon titles - is Dragon Quest. Although I'd also be interested to see how a similar model would apply to text adventures. That doesn't, of course, fit Twitch's setup, though.