NEW YORK—When Hillary Clinton closed a lively panel of women journalists covering authoritarian regimes in Russia, China, Turkey, and Georgia at the ninth annual Women in the World Summit in New York, Tina Brown, the event founder, summed it up with a new career suggestion for the former presidential candidate. “That was amazing, like brain sex. You’ve found your next career, Hillary,” she said, suggesting Clinton host a television talk show. “Brain Sex with Hillary Clinton.”