I don't think you can always say that if a given line of dialogue is in Trek, it was either for plot reasons or to provide insight into the characters. Sometimes episodes were a bit too short, and lines needed to be added to fill in time (hence the invention of "Piller filler." ) Sometimes, the need to have roles for all of the main cast in a week required the creation of "B stories" which were pretty lightweight. I do agree though that family would - in general - not be mentioned unless it was to provide insight into character or was plot critical.
As for Micheal Burnham, it was a great idea to make her bicultural - a human raised by Vulcans. It's probably the most interesting thing about her character. It would have been okay to have her be Spock's sister or massively alter the trajectory of Federation history by being blamed for - and ultimately stopping - a war with the Klingon Empire. My problem with DIS is it did both of them at the same time. While it's plausible that Spock never mentioned his sister, it's not plausible (yet - we'll have to see how Season 2 unfolds) that he wouldn't mention his sister if she was one of the most famous/infamous people in the Federation.
It's also yet another case of "small universe syndrome" - a toxic trend in Trek stories which has gotten worse as time goes on. I mean, wind back to TOS. Kirk was an important Starfleet captain - commanding one of the 12 constitution-class ships. He was important, but not the most important ever. Then the TOS movies had him save the Earth from destruction twice, along with help to save the Khitomer Accords. References to him in later Trek made him the most awesomesauce Captain EVAR. Picard's Enterprise wasn't just an important ship, it was The Flagship. Picard personally interacted with the Chancellor of the Klingon Empire (who knew his security officer) had the only Android in Starfleet on his bridge, and saved the Earth multiple times (including twice in the movies). I love DS9, but that show had ridiculous levels of small universe moments, with the titular heads of the Klingons, the Cardassians, the Dominion, and the Ferengi all visiting the station. Martok (and briefly Worf) ended up High Chancellor of the Klingon Empire, Rom ended up Grand Negus, and Sisko ended up a literal fucking god. VOY wisely mostly avoided this until the last episode, with the crew mostly a bunch of random schubs. But ENT created an awesomesauce captain we never had heard of before, who saves Earth from destruction in the third season. If the show wasn't canceled, we know he would have won the Romulan War and helped forge the Federation.
DIS looks at all this and says "hold my beer - I'll do even more in the first season!"
The Trek galaxy has thousands of inhabited planets, and uncounted numbers of people. It's completely implausible that our dear protagonists will always be at the fulcrum of galaxy-shaping events. I don't want a Trek crew to be comic book heroes - that's why I avoid comic-book media like the plague. I just want compelling characters who do their fucking jobs.
*exhales*