I wouldn't necessarily say it can be pinned down to qualities such as privilege or innocence so easily... Certainly, attractiveness plays a role, as does a certain level of arrogance (preferably justified) and competence- we like to see the strong, capable, proud characters suffer more than others I'd say, that's definitely true. But I used to be strongly into the full metal alchemist fandom (don't know if anyone else here is) and there used to be a LOT of fics written in which Roy Mustang got horribly hurt in various ways, and he wasn't a particularly privileged character and he certainly wasn't innocent, in any interpretation of the word- in fact, this has just reminded me of one other quality that these kind of "fandom punching bag" characters often have- they've often been shown to suffer in canon, or have a backstory which involves (or could conceivably involve) suffering. When it comes to Strange this is fulfilled by his wartime experiences and his grief after losing Arabella (particularly in the show which ephasised the drama of both events).

Come to think of it, I'm not entirely sure if there are any specific qualities of a character that can be pinned down when it comes to why people like to see them suffer... I dipped in and out of the BBC!Sherlock fandom for a while and it seemed to me as if people (typically different people but not always) were just as fond of hurting John in fic as they were Sherlock, despite those two having highly different personalities... I guess it's just a matter of personal taste, and in this fandom, Strange is one of the few characters who ticks enough boxes (whatever those are) to warrant this kind of treatment.