No they don't. The Chief Justice has additional powers, such as assigning the task of writing the opinion and administering the Court system. His rule includes more than simply deciding cases.
By contrast the appointment authority is a lifetime, exclusive power that he does not have to justify to anybody.
Ben, this is simply not true. He holds those powers at the pleasure of Congress, in contrast to his powers as a Supreme Court justice.
As for "lifetime, exclusive power", I think that an implicit qualifier of "under the normal course of affairs" is understood. Of course there are possibilities such as impeachment (yes, the members of the Supreme Court CAN be impeached), voluntary retirement, and change of laws. However there is no particular reason to expect it not to be for his lifetime.
And why is that relevant or useful in any way, since "the world" has no legal authority to influence the justices in any way (except for Congress passing new legislation after the fact)?
Furthermore the ability to write that reasoning is in itself the most important power of the Supreme Court. Because common law says that the reasoning in those cases can then be cited by any lawyer in any federal court in the country as binding precedent. And said precedent will be influential only to the extent that it makes sense, and the lawyer in question can supply an airtight argument that it applies to the case at hand. The more clearly the decision is written, the easier that lawyer's case will become, and the more power that the precedent is likely to have.
Therefore the reasoning that the justices produce for the outcomes that they want matters. A lot.
Not so.
Not exclusive because 8 other justices have equivalent authority on that ground.
Not unaccountable because he cannot exercise it without convincing 4 other justices to go along with him, through reasoning that is published for the world to see and analyze. (Which many do. In great detail.)
By contrast the appointment authority is a lifetime, exclusive power that he does not have to justify to anybody. No matter the legal underpinnings, there is no question that decisions made by his appointees shape the actual surveillance that takes place. The ability to reappoint every 7 years gives him real ability to, if he chooses, order judges to decide a particular way on threat of needing a new career. (It is doubtful that he operates in this way, but he can.)
There are limits on this power. It is granted by Congress. And so on. But it is the only power that he, by himself, can exercise with no oversight from anyone else.