Dave Johnson says his immediate reaction to his daughter’s diagnosis of Type 1 diabetes at age 4 was typical of any father’s. “It was my baby girl, and my response was I’m going to fix it or pay someone to fix it,” said Mr. Johnson, president and chief executive of the hotel management company Aimbridge Hospitality. “I dove in, spoke to some smart people. But then, I was hit between the eyes that there wasn’t a lot that I was going to be able to do.” That was 22 years ago, and what he did do...