I don't buy that Robin will be a proper successor to Batman. To be fair, I wouldn't have bought it no matter what they did. As far as I'm concerned, Bruce is THE Batman. I don't buy that anyone who loses their parents can just pick a costume and do what he did. The only successor I ever accepted was Terry McGinnis and possibly Dick Grayson in the comics, but Terry's circumstances were extraneous and Dick had all everything he needed justify his postion as Batman, such as life long training by him.
To be fair, JGL and Nolan wrote it as well as they could. Speaking objectively, they didn't really do a bad job. We see that the kid is smart, good hearted, has similar origins to Bruce, and already knows how to fight, and has similar morals. As far as potential successors go, this kid is as good as we could hope for. But it's just not the same. Bruce trained for atleast more than 12 years to be batman and nothing else. He trained with Ra's al Ghul in the League of Shadows, modern day ninja's. I can buy that Blake can hold his own. I can't buy that he's anywhere near batman's level, which is the level that's required to be expected to dress up in a costume and go fight gangs armed with guns.
And what about his resources? Wayne company is only starting back up from near bankruptcy and Lucius had all his R&D tech stolen by Bane. He has whats in the cave, but how is he going to resupply for long term? Can he even access everything that's in the cave? I'm sure Bruce would have stuff hidden, just in case. He is Batman, afterall. Or, hey, what his own finances? Bruce was insanely rich, so he could pretty much do what he wanted aside from the occasional obligatory party he had to keep up appearances. Where is this kid going to get money for food or housing? If he has a day job, where will he find the time to sleep?
And his allies? Gordon might back him, but the kid is alone otherwise. The first time he faced scarecrow, Bruce got burned alive and poisoned with fear gas. He needed Alfred to come get him and Lucius to make a formula for the cure. Alfred's off in retirement and he was only ever loyal to the Waynes. There is no reason for him to back Robin. Lucius might, but only if he is convinced that he was the replacement, which...how would he know? I don't think Bruce told anyone that Robin was his successor. Bruce needed those connections, otherwise he'd have either died after the first week or just gone entirely insane chasing down criminals and fighting other insane people. Rachel and Alfred and Gordon and Lucius were all necessary to make Batman work without him losing his humanity. Even if I accept everything else, who does Robin have to keep him safe at night?
Let me emphasize that I do think that they did as a good job with the idea as it was possible within this context. Bruce needed to stop being Batman, but Batman as a symbol had to remain, so a successor was needed. To make Robin a better fit than what he was would be implausible. It all just comes down to the fact that Batman was created not just because his parents were killed, but also because he was someone who was so broken by this one incident that few other people could believably have his kind of resolve to do what he did. He was also lucky to be born so rich that he had the finances to supply himself the way he did. And he had the loyal friends that were there when he needed them to be. A rare person with rare status surrounded by rare people. The stars aligned when Batman was born and Robin, good guy though he may be, is not Batman.