I've felt for a number of years now that Geoff Johns' best writing is behind him, and this book lends to that feeling, alas.
It's not that the book is terrible - compared to Scott Snyder for example this is disciplined and economical storytelling that won't leave any reader feeling ostracised from the book, but at its heart The Three Jokers is intellectually bankrupt. It has nothing to add to the story of Batman and the Joker, it just takes away. Diluting the two characters, not adding to them.
If Batman had deduced the Joker's identity from the very start for example how is it he could possibly have not noticed there was more than one man taking the name over the years? If the Joker is in fact not an anonymous enigma to Batman, and Bruce Wayne is well aware of his former identity and background as a failed comedian and husband and father, does it not take away any of the trepidation and doubt that Batman has whenever this most dangerous of his foes turns up again? Furthermore, if the Joker IS in fact the shattered reflection of this very feeble and not particularly bright failed comedian and unremarkable family man as presented, how is it this weak specimen transformed into a immensely clever and resourceful Psychopath, for that matter how is it those other two also became tactical geniuses able to match wits with the Batman, indeed why are all three so remarkably similar in looks and aptitude? It doesn't really make any sense when you look at it with any degree of thought...
All there is is an idea here, What if there were more than on Joker over the years, but you could apply that rhetoric to most longstanding villains - Lex Luthor might had several alternates for instance, Brainac as we know DID have numerous alternates! But while it worked for the ever changing Brainiac the idea is rather redundant when applied to a character like the Joker....
This is the kind of story you write if you want to shed a lot of prior continuity and refresh a character. They've done this with Thanos and Dr. Doom, where they say that prior incarnations in less popular stories were android doubles or clones. they even did it with the She-hulk, to nullify the X-Men issue where she got it on with the Juggernaut.
So, I suspect that part of the point here is to make a lot of the Joker's goofier appearances attributable to the Clown, now deceased. The problem is that DC isn't likely to forswear more goofy or problematic takes on the Joker in the future. We saw this same thing after The Killing Joke, where DC made no effort to keep the character at that level of quality. There was Death in the Family, with Joker running around the Middle East in a keffiyeh. We had the post Knightfall appearance, where he had long hair, a ball cap, dressed like a Miami tourist and acted like a movie producer, stuff like that.
This is a refresh point, but I doubt DC will make good use of it.