Why didn't the Librarian destroy the Composer? Or if not her someone else? The Forerunners had technology millions of years more advanced than us and they couldn't make it work. It's only possible function was to turn sentient life into mindless mechanical monsters. What possible role could it have to play in the future?
And why didn't she kill the Didact? By the time she imprisoned him he was more or less insane and beyond saving. What could he have had to teach the humans? If her plan called for a single person to remain behind to help the humans it should have been her and not the Didact. Or she should have seen to the salvation of several top Builders, Miners, and Lifeworkers to ensure that humans could reclaim the Mantle properly.
Finally, do we know what kind of upgrades the Librarian's AI gave to the Chief? He could hear the Didact's thoughts and was immune to the Composer's effects. I'm inclined to believe that his willingness to disobey orders to save the Earth and Cortana were also due to the changes she made to him.
Too lazy to answer most of that, so Master Chief disobeyed because he knew that if he didn't, humanity = Extinct. --- As he gazed upon his creation, he thought not of the success it would harvest, but of the deaths it would reap in the end - Me PM me for Friend Codes or GT
The Composer was still an option, just not perfected yet. The Promethian Knights, I presume, since they were composed Forerunners, most likely required the device. That, or it was nearly impossible to destroy. All the same, they don't give a reason, so we're left to speculate.
She didn't kill the Diadact because of two reasons. A) He was her husband. Surprise surprise, which leads to B) She had his sentence changed to imprisonment. He honestly acted in the best interests of the Forerunners, he was just wrong in his assumptions. Besides, do you get put to death for killing deer? Dogs? Lions? Nope. What he did was exterminate a possible threat to the Forerunners/Galaxy, which tore the "vote" per se. Thus, imprisonment, with the hope that he'd wake up and not be a total dick.
And his DNA changing made him disobey orders? Yeeeaaah, no... That's... That's pretty lame, honestly... His biology was simply enhanced a couple hundred years, thus making him immune. That's all it was. --- Royal Rose