My parents watch Criminal Minds and similar shows all the time, two friends of mine bought a book about a serial killer and my Twitter timeline is full of people talking about a new true crime podcast. It's just my bubble or is it a broad cultural phenomenon?
A lot of people are also living vicariously through the cops in the stories. Anyway, morbid curiosity should also draw them to stories of poisoners and assassins or bizarre happenings. Instead they zero on those who corrupt the rules of our society for their own pleasure.
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People either connect to the cops or the serial killer. Either way, they feel above the public.
I think that they don't really have to connect with either to enjoy a macabre voyeurism from it all.
I think people only really get into a piece of media if they can relate or connect to one of the characters.
I think people have a tendency to align themselves with or root for a side or character in a story, especially over time, but I definitely don't think they have to imagine themselves as one of the roles. Also, within your framework, you've left out people imagining themselves as the victims or would-be victims.
relating or connecting is not the same as projecting. I've played a little fast and lose with that so far, so I'll take some of the blame. However, you have raised an interesting point about the victims I've failed to consider. edit: forgot what the last thing I commented was, you make a good point.