Spoiler!!! Click to Read!:
Originally Posted by The Joker
Hang on, Bruce's mission and ideals never changed after Begins. He wanted to inspire Gotham's people. In TDK he was starting to see that was happening in a negative way. The copycats for starters ("That wasn't what I meant when I said I wanted to inspire people"). Then the Joker comes along, a character who was a total reaction to Batman's presence in Gotham, or as Nolan said "the logical response to Batman in Gotham". This was even hinted at in the Batman Begins ending when Gordon was talking about escalation;
Gordon: "What about escalation?"
Gordon: "We start carrying semi automatics. They buy automatics. We start wearing Kevlar, they buy armor piercing rounds"
Gordon: "And you're wearing a mask. Jumping off rooftops. Take this guy; armed robbery, double homicide, got a taste for the theatrical like you. Leaves a calling card"
Gordon is basically saying the underworld is going to react to the crackdown on them, and Joker is the reaction to Batman. This is another reason why he feels Harvey Dent is a better hero than Batman can be. A hero with a face who won't inspire the likes of the Joker ("I was meant to inspire good. Not madness. Not death").
Yes, being with Rachel was definitely something Bruce wanted. But it wasn't the defining factor in handing over Gotham to Harvey. He made Harvey prove himself and all by delivering him Lau and seeing could Harvey deliver the goods by getting him to sing. And he did. Harvey locked up half of the city's criminals in an afternoon. More of a triumph than Batman ever managed by himself.
When Joker started murdering people in Batman's name, he felt he couldn't endure that. Batman finally found what he thought was a limit Batman couldn't go passed. ("Today I found out what Batman can't do. He can't endure this").
In TDKR, he had succeeded in doing what he set out to do; saving Gotham and inspiring it's people. He did that via Harvey Dent by taking the blame for Harvey's crimes, and preserving Harvey's heroic image and the city's hope in Harvey's White Knight image. That resulted in the Dent Act, which essentially wiped out crime so much that Gordon was headed for a retirement.
So what else had Bruce got left? Just his grief over losing what he thought had been his only chance at true love and a normal life. Is it any wonder he was upset with Alfred for keeping such a big secret from him? She chose Harvey over him. Learning that harsh truth about Rachel was the kick up the butt he needed to move on and get over Rachel by realizing she had not been his once chance at true love. Hence why he was able to move on with Selina.
I'll put this simply. I know that Bruce's mission never changed. I agree with your post. Though, Rachel was still a part of it all from beginning to end.