In real life in something like that you need to re-establish a sense of normalcy and continuity. Which means you follow the laws, and the law said she was the next president to do succession. And Adama threw his support behind her because in a free and good society the military answers to civilian government.
Well, imagine you're in a foxhole, and you've got your weapon in your hand, and you turn around to the guy next to you, and it's a 4 1/2 foot tall 19 year old girl. Next, you find out she's your superior. Believe me- the first thing going through your mind would be-- "I am a dead man". "How do I get out of here".