The problem with killing Young and putting Rush in charge is that Rush is insane. Without someone to keep him in check, there is no possible direction for it to go other than him killing everyone For Science!
He isn't insane. He's actually the most consistently competent person in the whole show and the only one who has any consideration for the long term, which apparently means that he's a bad person by the show's logic.
No, he's insane. It's just that his insanity is masked by his rationality and reasonableness.
Really, he only cares about the Destiny. He doesn't want to hurt any of the other people there, he might even genuinely like some of them, but if push comes to shove he will always choose the Destiny over them. This was obvious in the very first episode.
When he saw his chance to get aboard Destiny slipping away he intentionally sabotaged and redirected the evacuation efforts. If not for Rush, everyone would be safe at home on Earth right now. He wanted to get to Destiny, everyone else be damned. He didn't care if any of them were hurt or killed. He didn't care that they would probably never go home again.
Or maybe he did care. But he didn't care enough to stop himself from stranding them on the edge of the known universe with insufficient equipment and no way home.
Rush isn't thinking about their long-term survival. He is thinking about Destiny's long-term survival. Right now those two mostly coincide. There will come a time when it doesn't, and when that time comes he will let them all die or even kill them all himself to protect Destiny.
