Last year, Primus Moore was among seven referees selected to officiate Oklahoma’s all-state high school football games.

“It’s inspiring for an 18-year-old kid to see that you’re 70 years old and you’re keeping up with them,” he says.

For Primus, being able to keep up with high school athletes was an even bigger deal because he was undergoing kidney cancer treatment at the time.

After his diagnosis in 2014, he had...