Link shows that when they introduced a program called "Zero hour PE" in schools (example), the kids get smarter.
“The object of Zero Hour was to determine whether working out before school gives these kids a boost in reading ability and in the rest of their subjects,” Ratey explains.
There are many other aspects of the overall program, of course, but the main thing to understand is that this is not the typical team-oriented curriculum. The emphasis is on fitness instead of sports. The kids are encouraged to find an activity they enjoy: Eighteen choices are offered, ranging from rock climbing to aerobic dance. The idea is to find something that allows a student to experience success. Grades are based on effort rather than skill. “Any kid who wanted an A could get an A, if he worked for it,” a teacher explained. “Any time you got a personal best, no matter what it was, you moved up a letter grade.”
The kids learned about getting fit—and how it would make them perform better in the classroom.
To make a long story short, it worked. The kids got fitter and smarter.
First, the kids in Zero Hour were sent off to their first period class in a “state of heightened awareness” and prepared to learn. At the end of the semester, they showed a 17 percent improvement in reading and comprehension, compared with a 10.7 percent improvement for students who opted to sleep in and take standard PE class.
Again, the emphasis is NOT on expensive team sports here for cheering on some abstract notion of a town mascot or honor vs a neighboring rival. It's on individual fitness, where it should be IMO to serve the most students.