SHANGHAI—Hundreds of Americans were preparing to fly out of Wuhan, bound for California, as fears grew at the epicenter of China’s health crisis. But more U.S. citizens aren’t leaving, having failed to secure a seat on the single U.S.-bound flight—or decided to ride out the emergency where they are.

A State Department evacuation flight promised relief for a segment of Wuhan’s roughly 1,000 Americans, as a lockdown triggered by a coronavirus outbreak turned the focus to the dangers of contagion and a long quarantine in China’s...