The problem with this prompt is that it could never be sensible. Even in a pitched battle sequence with the fate of the Galaxy on the line, the non-bridge crew (and I exclude I anyone in charge in Engineering ala Scotty or Geordi, or other "main characters") is going to be doing unimportant things. There's Klingons on the starboard bow, and you want to focus on the dude repairing the broken power conduit in jefferies tube 420, section 69? Okay, sure. And anytime there's not really anything going on, their usual jobs are going to be even more mind-numbing. As it turns out, most people have jobs that are rote and boring and would not make good television content. The only way this can actually function is if you have rare episodes as "filler", focusing on the mundane as a breather in between all the action. But an entire show of "filler" is impossible. Like a novel without a plot.
So the only way to solve this dilemma is to add in random wacky shenanigans that pretends to be important, when you already know it CAN'T be important.
What IS the genre? Because I feel the way I view sci-fi is very different from the way a lot of people do. I see it as a method of exploring the human condition using extreme situations that are impossible with our current universe and technology. A bunch of hot-shots blasting around the galaxy having super-amazing space battles is cool, but it doesn't explore anything. It doesn't ask the viewer anything but to sit back, let their eyes glaze over, and shovel popcorn down their throats.