Once onboard Destiny, Rush claimed he was put into charge. And later on Wray's superior told her to take command.
Which means nothing when the people with the guns believe they are in command, y'know?
Especially when the alternative is letting the IOA be in charge aka Stargate's council of useless gits designed to make the base commander character look good by rebelling against them.
Yeah, since IOA were only introduced to be thorns in the side of the SGC, trying to rehabilitate them into competent leaders sure didn't work very well.