The CJ decides who writes the opinion of the majority if in the majority, and the dissent if in the dissent. Its the job of the CJ to bring sides together in clear oppositions, and "horse trade" between bits and pieces of a decision so that its clear where a majority/minority lie.

The CJ's foremost political role is to ensure the judicial branch of government is seen as a politically legitimate institution which wields its power against the other branches in a constitutionally and poltiically legitimate way. If that slips, congress can start hiring/firing; and the executive, in the end, controls the guns -- they can be arrested.

To avoid being arrested or fired, the court has to keep all sides believing the rules they set are fair.

They have no power, in the end, but the power they are allowed to have. They govern by consent of the other branches, and that's trivial to take away