Two months ago, Kyle Vogt, the chief executive of Cruise, choked up as he recounted how a driver killed a 4-year-old girl in a stroller at a San Francisco intersection. “It barely made the news,” he said, pausing to collect himself. “Sorry. I get emotional.”

To make streets safer, he said in an interview, cities should embrace self-driving cars like those designed by Cruise, a subsidiary of General Motors. They do not get distracted, drowsy or drunk, he said, and being programmed to put safety first meant they could substantially reduce car-related fatalities.

Now Mr. Vogt’s driverless car company faces its own safety concerns as he contends with angry regulators, anxious employees and skepticism about his management and the viability of a business that he has often said will save lives while generating billions of dollars.

On Oct. 2, a car hit a woman in a San Francisco intersection and flung her into the path of one of Cruise’s driverless taxis. The Cruise car ran over her, briefly stopped, and then dragged her some 20 feet before pulling to the curb, causing severe injuries.