This doesn't make sense IMO - look at it this way:
Batman encounters the Joker and discovers his identity
It stands to reason that he finds out that the Joker's wife had died, sending him spiraling down into "Red Hood"..hood and into madness.
then, I see 2 options:
1.if the cover-up holds up under scrutiny, there should be no *reason* for Batman to look into a civilian death
or
2. it the cover-up raises suspicion enough to make Batman look into it, it should also catch the Joker's eye, especially if his first persona was like the Criminal's (which according to the story, it had been)
As for the Joker not caring, this story literally shows us that he *does* care and think about his "dead" family..
..which would make the cover-up quite shaky, and also connecting the "runaway" wife to the Joker, as well as basically exposing who the Joker was for at least several people..
Also, if that were true, the Joker would have done the same, torturing the Police officers for the information..