I have spent lots of time helping train my son's high school tennis team for 3 years. Yes, the first day of practice always involved running. Students would run about a mile after some warm ups. I don't recall anybody got cut because he ran a 8 min mile. Coaches pick up players mainly based on tennis skills.
I guess that if you are ranked in your section, you will likely make the team. Surely you need to improve your fitness including endurance if you want to make the varsity team and especially if you would like to play singles. You really want to run a mile in 7 mins. You would also need to continue tennis practice and to play a few tournaments before high school season starts.
As for running, my son had two coaches from Australia in the past 4-5 years. Both used running as a way of improving fitness. They made kids run multiple 400-500 M in about 30 mins with one min rest between runs. Kids hated these running when they were young
. But my son starts to appreciate the running because he is one of the fittest boys in our section. He started to run treadmill this winter to keep up the endurance training.