At least they tried.

If you're taking the setting seriously, yeah the command staff and bridge crew would not go on away missions, especially "exploring strange new worlds." they'd have a team of petty officers led by a Lieutenant for that.

On a TV show, even one with an ensemble cast, you still want to put your main characters in situations to explore those characters, to include the Captain.

It is my understanding that the West Wing was originally envisioned as having the President as a recurring role, not a main one. That it was supposed to mainly be about the West Wing staff and not the politicians themselves. Except Martin Sheen's Jeb Bartlett kicked way too much ass for that.

Possible solution? Make the exploration ship very small, with a crew of like 30 people, and then it makes sense that the main cast does everything.