After her kidneys failed from the same illness that took the lives of her mother and brother, Anne Rupp went on dialysis in May 2016, spending three hours a day, three times a week undergoing the blood-cleaning procedure. She hated it.

Rupp, who had polycystic kidney disease, joined more than 95,000 other Americans on kidney transplant lists. She knew the wait could stretch out for years.

But an experimental—and controversial—source of donated organs provided a far quicker resolution: Expensive medicines to treat hepatitis C have made it possible to use organs donated by victims of opioid overdoses who were infected with the once-deadly virus.

Six months after agreeing to be in a study in which patients in need of a kidney