I think if she tries to take control, the scientists will tend to side with her. They don't trust the military, and they don't trust Rush.
Well, at the beginning of the episode they did. That's why I think that Wray deliberately told everyone; she turned the scientists against Rush and they're already against Young.
As for the military folks, the person she should be courting is Eli. They've already shown that of the scientists/civilians, the military folks trust him most of all. Obviously not all the military personnel are content to follow Young. However, who would lead them? Scott is too devoted to Young, so he's out. Greer is a hothead, and moreover likely to side with Rush, since they're both impulsive and willing to take risks. TJ might be a possibility, but she hasn't really shown any willingness to lead nor break ranks. So instead of having two factions, she'd have to a) remove Young altogether, and b) win over the soldiers' trust with Eli.
In that case, then Young should pretty much say that Eli's job is to assist Rush in getting the ship up and operational (since Rush doesn't want to work with the other scientists, and none of them seem to be able to grow a pair and put up with him). Then tell Wray and Scott that Eli's also there to check up on what Rush isn't telling everyone else; they're the only two besides Young who really need to know what Eli's doing.
However, I still say that shouldn't be taking up all of his time and that he should do some of the physical training with everyone else. First of all, being in better shape is good for his mind as well; so it can help him with learning what he needs to know, and to be more alert as to what Rush is doing. Second, he volunteers to visit some of the worlds they're going to, but he really does need to get into better shape, just look at how he was straggling when they were at the desert planet.
On the second point, I don't disagree. Obviously he should be exercising for any number of reasons.
However, if Eli's job is to monitor Rush, the problem is Rush has been shown to work ridiculous hours. Keeping in mind Eli is not as experienced as Rush, it's probably all Eli can do to keep up with him; especially if Rush is trying to be covert in some of his actions. If I recall correctly, Eli even said he barely caught the planet deception.
What Young
should do is assign more people to go over the data. After all, Eli isn't the
only scientist on board, by far. That way it gets done faster and possibly more reliably. Of course, that goes back to the point I made above - Eli is the one the soldiers trust. So that's not likely to happen. Unfortunately, Young can't afford to confine Rush either, since he's by far the expert on Ancient tech.