I had a discussion with my friend about what makes stories compelling and interesting. His position boiled down to a single principle: relationships. He argued that the stories we enjoy the most are those that are inextricably tied to the relationships among the characters, whether they are former/current paramours, friends, rivals, etc.
I think that for some people compelling relationships are enough. I don't think that that is the full answer, and in particular I don't think that's a sufficient answer for everyone. If you look at books with broad appeal, they have a variety of features in common:
1) Sympathetic protagonists - You can have a story about an unsympathetic protagonist, but you'll lose a portion of your audience over it.
2) Compelling relationships - Your friend's criteria. Or to put it another way, the character loves and respects other characters, and does thing on their behalf, or hates or fears other characters and plots accordingly. You can eshew relationships, but you'll lose a portion of your audience over it.
3) Dramatic twists - Things happen in the story the reader didn't see coming. There are mysterious events. There are sudden obstacles and changes in fortune. There are scenes of tense conflict and danger. The readers perceptions of events are suddenly dramatically shifted.
4) Underlying Meaning - The conflict in the story is about something philosophically substantial. Like human relationships, this gives the story a timelessness so that regardless of setting, the reader feels something about the story applies to them.
5) Satisfying Conclusion - The story ends in a way that the reader is satisfied that it is complete. The conflicts in the story are resolved. The mysteries are explained. The main hindrance to the happiness of the protagonist has been removed.
6) Worth reflecting on - The events in the story ultimately made sense. Parts of the story that seemed unimportant are revealed as having greater importance than we at first thought. The questions that the story raised and the answers that it gave are worth chewing over.
I don't think that there is any one keystone. Instead, you build a story out of various pieces that people prize, and the more your stories satisfying those various needs, the more beloved it will tend to be.
I would say that some of the hardest parts of achieving this in an RPG is that the players have to be skillful as well. They are responsible for so much of that stuff, and if their input is missing or inept or unworthy, it doesn't matter what you do with the villains, setting, and chorus of the story. In fact, the meta-story of the RPG, whether we are talking about Knights of the Dinner Table, or The Gamers 2: Dorkness Rising, it's this conflict between the DM's desire to tell a great literary story and the player's ineptness or lack of interest in that as a goal or even perception that the DM's goal is orthogonal to their own is often the driver of conflict in the story.