Re: Why did the League of Shadows want to destroy a peaceful city?
I'm probably in the minority here, but I believe the League of Shadows did not return in TDKR. They were re-created by Bane and Talia with maybe a few ex-members.
As I wrote in another thread, the only member that we know died in BB was the leader, "Ra's Al Ghul". I believe Bruce was supposed to the successor to Ra's if anything were to happen to him. But since Bruce rejected the role and returned to Gotham, Ra's had to have picked someone else to take over if they didn't succeed.
Well, they didn't succeed but no one we know of from the League were arrested. So they either re-grouped and went back to their hideout or they disbanded.
So let's say the rest of the members returned home. Why would they agree to allow an ex-member (Talia) and one that was kicked out (Bane) to lead the group? Because her father died? I doubt it.
Also, the "LOS" in TDKR were only shown to be good with firearms. In BB, they were trained ninjas. There's a huge difference. I'm mean, Catwoman was kicking their buts on the rooftop until they started gun fire.
Again, I believe the "LOS" were disbanded after BB. While recruiting a few former members (how else would she find out who killed her father?) Talia and Bane re-created it "their way".
So now to answer the OP's question: Like Talia said, she was "finishing what her father started". Well, sort of. She cared about him (Ra's), but she really cared about Bane for saving her life. She was not going to abandon Bane as her father did to her. That's why she left the "LOS".
But deep down inside, it had more to do with torturing the person that killed her father (Batman). Her mother gets killed (for what reason, I don't know) right in front of her and now her father gets killed. She was not going to let that person get off easy. It just happened to be a coincidence that this person that killed her father (Batman) wanted to protect the city that was supposed to be destroyed. It was a perfect way to get back at Batman.