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?
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.