What you're describing is called a TSD Rallye. Unfortunately, and I DO mean unfortunately, there doesn't seem to be much interest in them anymore.
Back in the '70s , I was president of a sports car club in Columbus, Ohio which staged TSD rallye's several times a month. At that time there were either one or two other clubs in the area who were involved in the sport as well.
Would love to see a TSD club get started again. Those were great times for me.
.....sT