"The Brave and the Bold" paving the way for a new Batman other than Bruce Wayne would certainly be a first for the superhero's cinematic live-action projects, but it's certainly not an unexplored idea in other media. In fact, there's a ton of major instances of DC Comics rewriting the status quo to feature other Batmen as the protectors of Gotham City.
As mentioned prior, Dick Grayson holds the mantle as Damian Wayne's partner during his early stint as Robin. The anti-hero Azrael gets his start by taking over as Batman after Bane breaks Bruce's back in "Knightfall." Stories like "DCeased" have seen Damian himself become Batman when the time is right. There's even a longstanding element of modern Batman continuity in which Bruce creates Batman Incorporated, a force that employs numerous variations of Batman to protect the world. So on, and so on.
Even other "Batman" adaptations have followed this route, with the animated series "Batman Beyond" being entirely predicated on Bruce handing over the mantle to Terry McGinnis and the movie "Justice League: The Flashpoint Paradox" exploring an alternate reality in which Bruce's father Thomas is Batman. Heck, Christopher Nolan's live-action flick "The Dark Knight Rises" teases Joseph Gordon Levitt's character becoming the new Batman at the end. In any case, exploring the DC Universe through the lens of a non-Bruce Batman is less of a far-fetched idea than some might think, and it could be exactly what this version of the Batman mythos needs to distinguish itself from past incarnations.