You got some of it wrong, in fact: The gang that existed while Batman was retired was the Mutants, a completely horrible gang of sociopaths and sadists (Miller

You got some of it wrong, in fact: The gang that existed while Batman was retired was the Mutants, a completely horrible gang of sociopaths and sadists (Miller is not one for subtlety) lead by a monstrous beast of a man (again, Miller not subtle). Their actions force Batman out of retirement and, once Batman defeats the Mutant gang leader in front of his gang, certain elements of the gang restyle themselves as Sons of the Bat and go after other criminals. Again, though, they're still horrible violent sociopaths and sadists, and Batman has to spend some time cracking their heads together. So you got a number of the main points right.