It doesn't have to be 'either/or'.
The DCs had daily PE classes of 45 mins each in high school in the US, with various required PE classes necessary over the course of the four years, including movement/dancing, team sports, racquet sports, gymnastics, self defence, fitness (incl how to use fitness machines, weight machines and safety in the weight and fitness gym) swimming, adventure ed (canoeing in the pool, safety, rock climbing, etc). You would end up repeating some areas, or you might end up playing soccer and then basketball for team sports, or tennis and then badminton...
On top of that they did sport practice both before and after school during the season of their sport (before school 5 to 7.30 am with classes starting at 8 and afterwards from 3.15 to 5.30 with a meet following about three times a week when the season got under way). DD1 got a PE waiver in her final year as she had to take a certain art class to get her portfolio filled.
DD1 did swimming (winter sport) and water polo (spring sport) for her final two years, and did badminton for her first two years (spring sport). The other DCs did football (DS, autumn sport, with training lasting through the preceding summer and weight lifting required in the winter), and badminton (spring sport, DD2 and DD3). Swimming was very much as Pagwatch described. Football consisted of constant drills and strength / speed / stamina training. Badminton was a matter of racquet drills/ work on form, stamina and footwork. Nobody was expected to bust a gut/ break an ankle during training.
For extracurricular team sports there was a C to play rule -- if any of your grades fell below a C you were benched until you brought it back up to a C average. Any infraction of underage drinking or smoking or narcotics laws (including references to drinking etc, on FB or other social media) meant you were kicked off the team.
Using left and right sides of the brain makes the whole brain work better. You can do this with art and music and sport if the sport programme is well thought out. Pushing yourself to get up for those early morning sessions in the pool or on the track or in the weight room builds character and makes you into a more determined student. Organising your time so that you get all the homework done on top of the commitment to sport is a skill that develops maturity.