I care about the show because it’s fun and I enjoy watching it, but I don’t take it “seriously”; it’s fiction delivered episode by episode, I can’t really take it any more or less seriously that Game of Thrones or Frasier. There’s stuff I’d like to see done more or done better or avoided, but nothing that really irks me that much.
As for other fans, goes the same as any other fandom, I guess. Look at trekkies or SW fans; some spends hours arguing about inane details, or drawing diagrams for some plot-irrelevant prop seen in two episodes, or whatever. Same with Supernatural or Doctor Who or whatever other show runs these days as well. The loud ones and the extremes stand out, the “rational” silent majority is largely ignored.