Open this photo in gallery: U.S. President Joe Biden speaks during a news conference on the final day of the NATO summit in Madrid, on June 30, 2022.Susan Walsh/The Associated Press Mark Leonard is director of the European Council on Foreign Relations. When NATO leaders descend on Vilnius this month for the alliance’s annual summit, they will demonstrate that the organization, newly united behind support for Ukraine, is far from “brain dead,” as French President Emmanuel Macron infamously described it in 2019. But NATO’s new vitality belies a larger problem: The West’s failure to convince the rest of the world...