Yeah, I think I’d argue that the Pre-Flashpoint BatBooks were a great example of how an appearance of redundancy or “purposeless-ness” would be disproven simply by having great creators hitting their strides on a lot of characters and books all at once. Dick and Bruce, in costume, depended on viewers identifying costume details to tell them apart on a shallow level… but that wasn’t a problem, because the entire fanbase was engaged in the stories on a much deeper level.
Besides, I think the Red Robin role’s main “purpose” for Tim at the time was actually about making sure he wasn’t overlapping with Damian, rather than finding Bruce.