I was looking for Fiction that felt like Star Trek earlier, and I came up with very little.
Because Star Trek is a synthesis of serious War sci-fi with a bunch of soap opera conventions.
The things that have the big ship with the large crew and the hyper-competence tend to be barely disguised fascist propaganda.
The fiction that does the most character driven stuff, tend to feature very small crews of people who are just barely scraping by. Sometimes this is really lovely (The Wayfarer's series, for example) but it is also very different.