You can certainly do smaller, more personal stories on TREK, but when the whole idea is that you're out there exploring strange new worlds and civilizations, and crossing vast vistas of space, that sort of lends itself to storytelling on a similarly impressive scale. Space is big, as Douglas Adams wrote, so sci-fi tends to big stories as well.
Granted, you don't have to save the entire galaxy every episode, and I suppose it would be possible to tell a story like "Journey to Babel" or "All Our Yesterdays" from the POV of a junior officer, but, again, if Ensign Foley is saving the ship and making the tough calls week after week, audiences may start rolling their eyes and wondering just how useless the captain is anyway.