‘Obama’s lean years are over and we are now witnessing a new chapter which we hope will be an alternative to the mistakes of the abysmal past.” A Trump supporter’s postelection boast? Hardly. It’s how Turki Aldakhil, a Saudi broadcaster who is general manager of the Al Arabiya news service, greeted President-elect Trump in a recent article.

Some American observers fear that Mr. Trump’s comments on the alleged danger from Muslim immigrants cripple his prospects for effective diplomacy in the Middle East. But at the mid-November...