This is an interesting take, especially since I've been reading Caroll Quigley's The Evolution of Civilizations. Prof Quigley had a similar observation on the evolution of American football as an institution. Quigley described football originating as an impromptu intramural program to keep students exercising. Over time, football naturally became an organized club sport and spectator-funded as people focused on winning the game against other groups of people (clubs, colleges, cities, etc.) The natural institutionalization of American football led to the NFL and NCAA today, where the athletes are getting more exercise than they need, while the rest of us who need the exercise are, instead, watching from the stands, sitting down and converting beer into calories. The institutionalization of football has made football an abject failure at achieving its original purpose: Instilling habits of exercise among the youths of this country. Prof Quigley arrived at this conclusion back in 1961. He saw all the other college sports programs in the same light, but football and basketball were the most prominent offenders.
Organized team sports definitely are not helpful in promoting general athleticism among the general population, speaking as an uncoordinated nerd myself :) By its very nature, team sports seek to exclude. The very structure of a tournament competition incentivizes the teams to seek the best players, and exclude the general peons.
It is a problem found in most organized competitions. Math teams and Academic Decathlon, for example, both seek to promote scholastic skills among the student population. However, the team competition format discourages the less practiced students from participating. The tournament format of Academic Decathlon additionally leaves most participating school teams with little to do for much of the year. [Academic Decathlon has a regional meet in Nov, a state meet in Jan/Feb, and a national meet in Apr/May. One to two teams per state go on to national.]
At this point, though, organized team sports are here to stay. With the entrenched institutions and interests of football, basketball, et al, in the US and around the world, reform is near impossible. As Prof Quigley would say, we'll have to try circumvention. The Japanese morning calisthenics program, or the Chinese morning Taichi, both look pretty good from here.