I'm not really sure what the point that of non-competetiveness is, especially in schools. Getting that competitive spirit in fairly early and fostering that idea to be the best early on is going to prepare them more for the adult world than saying "it's ok honey, you were all equally bad in the end so it didn't matter anyway"
By the very definition of Sports, it is competitive. Removing the competitive aspect to sports removes the idea of sport altogether, and by extension it rules out alot of potential sports that can be played.
In reality competition is always there, by removing this competition you soften the children and remove them from the realities of life itself. Life is a rat race, and friendly competition is the best place to learn this.
Competition also spurns the spirit of the children, it drives them to do better and try harder than they have done before.
When children leave school or even throughout, competition is a prominent aspect of every persons life. Through creating non-competitive school sports days, especially in primary schools, you are stopping children gaining important experience.
Whilst yes it does protect some of the less athletic children's feelings, and encourages sport to be fun and interactive for all, it takes away the point of a sports day.
Sports days are a chance for the less academic children to shine, showing off their athletic ability instead, and through hosting a range of sports and allowing children to pick where they take part, children will only compete where they are comfortable doing so.
In turn sports days should be about teaching good sportsmanship, rewarding students for their achievements, and most of all teaching children that just because you are not the best at something it isn't the end of the world. Your team might not have won, but you tried your hardest and you enjoyed it. As long as children compete in fairly matched ability groups, don't put the quickest child in the school against those who can't run as well, then no child is going to spend the rest of their life dwelling on why the team they were in didn't win at tug of war, or why someone beat them in a sac race.
These non-competitive school sports days are shielding children's feelings at a time they can learn to bounce back from defeat and try harder next time. This only sets children up so when they fail in the future, which they inevitably will at something, they don't know how to handle their feelings or move on and take it as a learning experience.