Out of all of Batman's Robins, he failed Jason Todd the most tragically, but it's for that precise reason that Red Hood would actually make a surprisingly capable version of the Dark Knight. After Dick Grayson moved on from being Robin to become Nightwing, Batman needed a new sidekick. He eventually recruited Jason Todd after the boy was caught trying to steal the tires from the Batmobile. Sensing Jason's troubled side, Batman tried to focus his trauma and anger into vigilantism as had worked for Dick. But Bruce Wayne quickly realized that Jason was a very different kind of troubled youth. He struggled to contain Jason's bouts of violence, which originated from his traumatic and abusive childhood, and even suspected Jason of murder when he was Robin.
Given Jason's abusive past growing up, Batman made a grave error bringing the boy to horrific crime scenes. Batman could relate to children of trauma, but he didn't revisit this trauma on a nightly basis as a superhero. Bruce should have known that Jason needed time to process what he went through. Being around murder victims was not the best way for Jason to grow into a healthy hero. In fact, it probably drove Jason's erratic behavior and set him on the path to inevitably becoming the Red Hood.
But his time both as a child of abuse, then a criminal, and also a Robin means that Jason is uniquely suited to be Batman and nurture other children in his position. He understands better than any other Robin what it means to grow up with trauma and to have to relive that trauma every time he puts on the uniform. In fact, as the Red Hood, Jason has made it his personal mission to protect those that can't protect themselves more than any other of Batman's protégés. This stems from his own personal experience of being victimized without anyone to stand up for him. A great example of this is in Batman: Urban Legends #1 from writer Chip Zdarsky and Eddy Barrows, where Red Hood takes a boy named Tyler under his protection after his mother dies from an overdose, just as Jason's did.
Red Hood has proved that despite his initial brutality towards criminals, his true goal is to protect and nurture the innocent. While some fans think of Batman as a solo hero, rescuing and tutoring his proteges has been a longtime priority for Bruce Wayne, who has raised a veritable army of young vigilantes. It's in this aspect of the Batman mantle that Red Hood has proved he'd excel, as his focus is squarely on helping those in need without judgement or condemnation. Jason's own childhood and his later failings mean that unlike Bruce, he's not looking to hold anyone to a near-impossible standard, just to help them survive their grief.
One of the major questions hanging over Bruce's time as Batman is if he actually helped those he took under his wing, or just guaranteed a future of violence and conflict. Should Jason Todd take on his mentor's role, his focus would be on the former, giving Gotham a Dark Knight who, while admittedly brutal, places the highest priority on the victims of violence rather than the perpetrators. With a semi-heroic new role in DC continuity, Red Hood is closer to Batman than he's been in years, and has unique qualifications to eventually become the Dark Knight.