Of course he would. Of course he'd leave the Cave to someone capable, someone who'd want (and would wear) the mantle well. Hadn't he tried to do something similar in the real world, when he'd 'retired' to focus on the trial?
"Gotham survived?"
It was a fictional city. Bruce didn't know why he was so worried about its fate - it wasn't really his home, and it didn't exist to be protected. He ought to have been more worried about California, but the truth was that he felt a tie to that awful, imaginary place. He cared about what happened to Gotham and its people.