SANTIAGO, Chile—An annual summit of Chile’s business elite has typically celebrated this country’s economic success by gathering some 500 titans of industry for a congratulatory speech from the country’s president. This year, as looters and protesters have taken aim at Chile’s establishment, the group convened only 120 attendees and had a psychiatrist address them instead. The gathering’s title: “Day of Internal Reflection.”

Bernardo Larraín Matte, who heads the Manufacturing Development Society that holds the summit, sounded...