Worth pointing out that for most of original Robin's time in the costume (1940-1984), Batman wasn't depicted as a dark, brooding, grim, gritty character. Sure there was some violence, but not like we associate today with the character. That didn't really happen until the 70's, and that's when we also saw Grayson move away from Batman. Then from '83 to '88 we have Jason Todd. Post-crisis Jason is a darker take on Robin. He killed a dude once as Robin and was like, "Meh." By the time Tim comes along, we see Batman accepting Robin, but also trying to sideline him and keep him out of the action as much as possible. Then we come to Damien and that murderous psycho is great, but it's hard to argue that he's a feel-good, plucky hero who needs to be sheltered from the world.
Tldr: The past 40 years for Batman have been very different from his first 40 years. Robin made sense in the first 40, and in the second half of Batman's life, the relationship has been altered to reflect the new tone.