ATSB investigate aircraft accidents withing CASA's registrating and licnsing system, and can investigate a self administering sports aircraft organisation in certain circumstances. They are an independent body as you say.
CASA is a safety body, and ATSB do the investigations for them as outlined above.
The State Police investigate accidents on the road, in workplaces and generally and provide a Brief to the State Coroner.
Self Administering Organisations, if they are smart, investigate their own accidents which go into their own databases to help reduce the accident rate, and also assist Police where called on with their specialist advice, for the Coroner's brief and the Coroner makes a finding on the deceased plus may make recommendations for a Sport/Industry etc.
A Self Administering Body is just that. It can sit there with its tongue in the sude of its mouth and just cop the lawsuits without doing any investigation at all. However it's much smarter to pass its own safety regulations, administer its one member standards and aircraft standards, investigate promptly when something goes wrong and take action to minimise any repeats.