(Note: I’m looking at the history of stories told in the DCU, not just New 52 canon)
There are several cases you could make for the other Robins: Dick Grayson's relationship with Bruce and evolution to Nightwing, Tim Drake's deducing the world's greatest detective's best-kept secret, Damian's unparalleled training and possible birthright. While these are all valid points for characters who all brought something to the mythos, I think they all lack a core detail—the darkness. Bruce Wayne is many things, but so long as he wears the cowl, it's impossible to say that there isn’t at least a little darkness at the core of his character. It's the death of his parents that’s driven him all these years against both his classic and most recent villains. I believe it is this darkness that could have allowed Jason Todd to be the Batman's greatest ally, because in many ways, it made him Bruce Wayne.
Now that may sound a bit bold, especially when you consider the similarities in Bruce and Dick's origin stories. Both saw their parents die right before their youthful eyes. That's got to be a tie that binds. Although it may seem valid on the surface, I think the way Dick notably conducted himself as Robin refutes the possibility of the two empathizing over their shared loss. Dick is a circus performer at heart. As such, he was infamous for bringing theatrics to the role of Robin. Where Bruce is a wrecking ball of vengeance, Dick is a spring-loaded butterfly. Even if Bruce had found someone to mentor him through his struggle as he did Dick, I don’t think he would have been smiling and somersaulting his pain away as Dick did. It's just not who he is. This distinction further illuminates that Jason Todd is Bruce Wayne.
Jason also lost his parents and descended into darkness. I believe his story may have been Bruce’s had the young master Wayne not been privy to the comforts of an unimaginable inheritance and a butler always on standby to provide him with unconditional positive regard. I believe that by forcing Bruce to try and console and save Jason, he was forced to confront himself. That period of Bruce's life is what drives him. Having a character like Jason—who so perfectly embodies the toll that Bruce’s pain could take on a life—wield the mantle of Robin led to stories that simply would not have been possible with the others.
Sure, it is granted that I'm basing the case for the best Robin off what they could do for Bruce, but hasn't that always been the justification for a Robin? Tim and Dick were the lights in Bruce's life that kept him from going too far, and Damien... well, let’s not go there. By the time Bruce got to Jason, Dick had finished his tenure as Robin and been the positive force Bruce needed. This could have left Bruce and Jason in a position to go down the rabbit hole together to conquer their demons. There's a clarity that comes from helping others with a similar issue that you just can't come to on your own.
Bruce has all the gadgets, all the money, all the training and all the experience to make facing external threats a thrilling story. Much like Superman, however, his true challenge comes with confronting his own darkness each and every day. This is why Jason was, and remains, the perfect sidekick for Batman. Bruce Wayne will never face a villain worse than his past, and no Robin had ever been better tailored to help him face it than Jason.