Despised by some, admired by others, the U.S. has been the Middle East’s principal power for decades, providing its allies with guidance and protection.

Now, however, with Russia and Iran thrusting themselves boldly into the region’s affairs, that special role seems to be melting away. As seasoned politicians and diplomats survey the mayhem, they struggle to recall a moment when America counted for so little in the Middle East—and when it was held in such contempt, by friend and foe alike.

“It’s...