Also it makes just as much sense like this: 'sports is to sex as pornography is to war.'

By adoption of sports, do you mean universal participation or spectation?

The former I suppose would be a net positive, if you limit the requirement to just making sure everyone does at least some physical activity every day, but if you mean that everyone should be required to play some sport as a significant pastime the negatives start reducing the net positive.

If the latter, I must vehemently disagree. We would probably be better off banning spectator sports entirely rather than making everyone watch.