If it isn't clear, what I mean by a fan-insert story is a story about a fan falling into the world of their fandom and interacting with their favourite characters.

Ideally, I think this works best in a fandom which already includes fourth wall breakage. It can also work in a story which includes magic or fantasy, since these elements can explain why/how they end up falling into the story.

I also think that, for it to work, the author needs to be very realistic about how a fan would react in such a scenario. They would miss their old life. They ought to be frightened when they realise that the dangers within the story are now real, and notice the down-sides to the characters' lives. They shouldn't automatically already have all the skills necessary for their new in-story life, or gain those skills suspiciously quickly once they start to train.

If you're going to write a fan-insert/canon-character romance, it works best as a slow burn. Maybe they should become friends initially. Or maybe the insert character should change their mind about their favourite character upon having actually met them. All these help avoid it being a meet-favourite-character-and-instant-love scenario.

The vast majority of fan insert stories are of the "run a mile" variety but I think they can occasionally work.