No, he doesn't. This is a directly, overtly inaccurate claim.

John Roberts real exclusive, unaccountable, lifetime powers come from Article III of the US Constitution, which within 50 years of the ratification of the Constitution came to give the Supreme Court the power to review and overturn the laws of Congress and to issue injunctions against presidential administrations.

Ezra Klein in this article isn't referring to Article III courts. He's referring to the FISC, the FISA courts, which aren't Article III courts. Judges on the FISA courts, unlike Article III judges, don't have lifetime tenure, can't hear adversarial cases, and have jurisdiction over a single class of controversies: foreign surveillance requiring cooperation from US entities or which could collect information about US citizens.

The key distinction between FISC and the Article III courts is that FISC is a creation of Congress. The Constitution is silent on the issue of foreign surveillance. Contrary to popular opinion, the Fourth Amendment does not enjoin the US from spying on foreign powers; no industrialized nation in the world (that I can find; correction welcome!) enjoins itself from spying.

Without specific guidance from the Constitution, the rules for spying are simple: Congress passes laws, the President enforces them, and the Supreme Court overturns them if those laws conflict with the Constitution. That is the exclusive power to shape the surveillance state.

What Congress did with FISA was insert an administrative check on the NSA. They chose extremely confusing terminology and a confusing structure to do this with; they created a "court", appointed by the Chief Justice, to conduct internal hearings on the legitimacy of individual surveillance efforts. But that "court" functions unlike any other court in the US, because it isn't a court; it's a review board that happens to be staffed by judges.

John Roberts only authority with that court is to appoint justices, who must come from the federal court system. So far as I can tell, he has no other authority; he cannot, for instance, recall FISC judges.

But, more importantly, if enough Senators or Congresspeople decided to alter the way that FISC worked, John Roberts would have little ability to stop them. Congress could probably ban all foreign surveillance (that might create a Constitutional crisis, since foreign spying is a war power allocated to the Executive). They could certainly pass laws that would punt cases to Article III courts. They simply choose not to.

There is nothing Congress can do to eliminate judicial review. John Roberts has a lifetime power to persuade SCOTUS judges to overturn the laws of Congress. That power is unaccountable; if anyone in the US Government decides that SCOTUS is abusing judicial review (and many people frequently do decide that, which is what they mean when they say the courts are "legislating from the bench"), there's nothing they can do except to ensure that the next judges appointed to the courts are more congenial to their views. And SCOTUS's judicial review powers are exclusive; the President can veto a bill, but Congress can override that veto. Only SCOTUS has the power to take a law that has passed and strike it down. Roberts FISC authority, and the authority of the FISC judges themselves, are nothing like this.

If citizens want to challenge the surveillance state (and they should), they have all the same means to do so as they do in civil rights cases. They can lobby to have laws passed. They can bring suit, and, if they can show that their Fourth Amendment rights are being abused (or, in Google's case, their First Amendment rights), they can have the government enjoined from abusing them. The immediate response to this will be to point out how hard that is to do, but it's as hard as it is with any other civil liberties controversy, and none of the difficulty comes from John Roberts ability to point FISC judges.