What are the public and private rights resulting from the driver who caused an accident that led to my brother's death, and who has not been held accountable after three years? And is it permissible for the family to waive the blood money without referring to the court?

As for the public right of the state, there is no capital punishment for accidental killing, unless it resulted from recklessness. In such a case, the state may punish the killer with a deterrent penalty it deems appropriate. As for the private right, it is the blood money (diyya), which is borne by the 'Aqila (male relatives on the father's side) and deferred over three years. If the perpetrator has no 'Aqila, then it is taken from the public treasury (Bayt al-Mal). If that is not possible, then it is taken from the perpetrator's own wealth. As for forgiving the blood money, it is a recommended act in Sharia, but conciliation should be taken into consideration. If signs of repentance appear on the killer, and he is in need, then forgiveness is preferable. Otherwise, it is better to take the blood money so that he may be deterred.