At my high school, sports were required. It was considered an important part of the educational process. You didn't have to be good, but you had to do something. I think that you'd improve test scores more with healthy bodies than with more time in classrooms. Sports should not be for just the top athletes. Health is important for everyone.
It is, but that's why physical education should be compulsory. Sports should not be. I don't like schools dictating what people can do in their free time, and it's a pretty inconsiderate imposition to people who are moderately-to-extremely introverted. You're probably causing them undue stress.
We didn't have PE. We had the sports requirement instead, but note that it didn't have to be at the varsity or JV level; there were "club" sports, too, (which was what we called intramurals, and not to be confused with the types of club sports being discussed by others on this thread). With everyone participating, there were a very wide variety of sports, including a wide diversity of both team oriented ones like football, water polo, hockey, or lacrosse, and more individual ones like golf, tennis, or cross country.
There were certainly places where introverts could find a home. I think they were more comfortable in this environment than they would have been with a PE requirement. In PE you are around many people, and have to participate in a broad range of activities, many involving other people, and many that a non-athlete might struggle at. With our sports requirement, you could find some sport you felt comfortable at, and enjoyed, and focus solely on that one.