LOS ANGELES—Even before breaking for commercials, the roomful of celebrities and Hollywood executives at the 75th Golden Globes began writing the script on an “Oprah for President” campaign.

The billionaire talk-show host and media mogul had just finished a thundering speech on Sunday night accepting the Cecil B. DeMille Award for career achievement—an eight-minute oratory that wove together childhood anecdotes, racial-justice themes and the country’s unique political moment.

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