Re: Why did the League of Shadows want to destroy a peaceful city?
The League of Shadows are dogmatists. Ra's had set his sights on destroying Gotham for many years, and had tried twice in two different decades under very different circumstances. I don't think they saw Gotham's attempt to fix itself in the wake of Harvey Dent's death as any different from when it tried to fix itself in the wake of Thomas Wayne's death. In fact, Gotham becoming a police state with an ever-widening gap between the haves and have nots was probably just seen as further insult to injury. They don't think too highly of politicians on principal, hence their existence as a global organization working from the shadows.
Bruce- "This man should be tried"
Ra's- By who? Corrupt bureaucrats?"
So yeah, Bane finding that letter only vindicated the fervent beliefs that were already driving the mission, as Agent Orange pointed out.
In TDKR, Gotham is a city well past its expiration date in the eyes of the LOS. But with Bane and Talia at the head, there's a deadly combination of hatred, vengeance and brutality mixed in with fulfilling an old mission.