Just why send bridge crew. Those crew are important and mean to stay on the ship. Unless the mission required someone on the top like first-mate, you should send your boarding team or land team on mission.
Most of the eps, I see a dangerous mission with 2-5 bridge crew member. Sometime it is include both captain and first mate.
Sometime, it is fighting sorties, a first mate said: I need volunteer who is trained in close combat. And multiple bridge crew go with him. The next scene is the crews start shoting handgun in some dangerous place, maybe they also teleport inside enermy ship (boarding action), and sometime is fighting kungfu.
Why not send a sergeant and handful of soldiers ? The ship is quite big but there are no spare personel to send ?
Now you can judge someone’s social status by whether they drive a luxury car, if they live in a mansion, or how many employees work for them. In a post-scarcity society, people who want high status have to earn it. By going down to unexplored planets themselves, the bridge crew get to claim any new discoveries as their own. People all over the galaxy will hear about who made first contact with a new civilization.