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 ?
Because the crew sees itself as researchers, scientists and diplomats. There is no reason to have a hazard team. We can assume that the encounters we see on screen are only the unusual ones…