Yes I agree that Joker works better as a mystery...BUT it's not like his character has never had a back-story given to him before. Is Jack Nicholson's Joker any less the Joker because Burton gave him a backstory? He's the Joker I grew up with, and I still love him to this day. Heck Mark Hamill's animated Joker was also given a back-story of sorts in Mask of the Phantasm.
Besides what exactly did we learn about Jerome...that he grew up at the circus and killed his mother? That's not a lot of info really.
And IF Jerome does become the Joker...whose to say that in the future when he does take the plunge into the acid and becomes the killer clown, that the police will figure out that he's the same kid who terrorized them years before?
And even if they did... how would that info help them?
Example:
Gordon: Jerome?
Joker: Jerome is dead, you can call me Joker!
I know I stole from Nicholson, but Jerome sort of reminds me of Nicholson's Joker more-so then Ledger's Joker.

t:
But since he was always a psycho it's not like they could use his 'past life' against him anyway.