In 2013, as Theranos Inc. was in the throes of trying to get its blood-testing machines into a national drugstore retailer, Adam Rosendorff walked into the CEO’s office and made a request: Wait.

Dr. Rosendorff, then Theranos’s lab director, felt some lab tests were too unreliable to use on real patients. He asked Elizabeth Holmes, founder and chief executive officer, to delay the public launch, according to court testimony.