Okay, guess I get your point. I suppose my dad is kinda like what you describe... watches (and reads) a lot of things, but doesn't really get into deeper analysis of them, nor volunteer his thoughts on them. Although if someone else gets a conversation going, he can hold his own.
He's kind of an outlier, though. I don't see my extended family too often these days, but I did visit for the holidays last year, and a bunch of us went to see the latest Star Wars film. Now, as I have mentioned around here before, I am not a Star Wars fan. I don't give two hoots about its continuity, I was literally unaware (until informed by posters here) that there was an animated TV show, and I wouldn't even bother to see the movies if friends or family didn't make a point of it. Nonetheless, in the aftermath of seeing it, we all had a very spirited conversation about the strengths and weaknesses of the movie, and various points of plot and theme and characterization, and questionable bits of story logic. Nobody was bothered by Luke's grumpiness (I really think only a certain faction of die-hard SW fans care about that, the kind who idealized Luke as a role model despite all on-screen evidence), but we all (including my dad) discussed lots of other aspects of it.
My girlfriend is Chinese. She didn't grow up in the US, so she had never seen the original SW trilogy. Therefore (at her behest) we watched it together last fall, before TLJ came out. She guessed in advance that Vader was Luke's father (among other plot points). SW has never been one for subtle, intricate plotting.
I think movies in general (especially movies that aren't part of ongoing franchises) might be a better example of what we're talking about than TV shows, because people don't go into them with lots of pre-formed expectations and loyalties. Even so, when I see a movie with friends, a big part of the experience is dissecting that sucker afterward — its strengths, its weaknesses, everything about it. That's why it's usually more fun to share a viewing experience with people than to watch something by yourself — for the conversation afterwards! Sometimes the conversation can be more interesting than the movie, frankly.
And when it comes to TV shows that do build up viewer attachment over time, that's just amplified. Not just Trek; take any show that has enough backstory and worldbuilding, or intricate plotting, or nuanced themes, to make for a meaty discussion... from Game of Thrones to Doctor Who to House of Cards to, heck, Rick & Morty... and by god, we discuss it! In depth. That's part of the fun. If I don't have friends to watch something with, I often go to online forums like these (but less specialized... the comments sections on AV Club reviews are usually insightful, for instance) to find that kind of conversation. Other people's perspectives often inform my own in unexpected and fascinating ways.
I don't think of this as "obsessive"... I just think of it as engaged. If a show isn't complex enough to engage me on that level, I simply tend not to watch it. (Helps keep my time spent staring at glowing rectangles down to a reasonable level.)
Basically, I analyze everything in my life, including entertainment. I hang out with people who like to do the same (if not always to the same degree as me). Perhaps it's an artifact of living in a college town, I dunno. But that's why I'm always surprised if people don't do that... and why I'm curious (as I asked) about what other people did react to in a show like DSC, especially if it's different from what I reacted to.