DUBAI—Billionaire Saudi Prince al-Waleed bin Talal has returned as the kingdom’s public face for global investors, emerging a year after his detention to help the royal family survive the crisis over journalist Jamal Khashoggi’s killing.

Prince al-Waleed was the best known internationally of the Saudi businessmen, government officials and royals detained last year in what Saudi authorities called a corruption crackdown. Prince al-Waleed wasn’t publicly charged, but his three-month imprisonment raised doubts about his quasi-diplomatic...