Yes, non-competitive school sports day every day is the way to go. That way young people won't have their confidence damaged before entering the workforce.
I still find it odd that the Government refuses to learn from Private Schools.
My daughter's school is fiercely competitive - the girls are encouraged to play competitive sports, and the best to play against other schools. At the same time they have moved away from forcing people to be miserable when doing PE and encourage team work and sportsmanship (sportswomenship?). For example, the best players in the first 11 for netball will act as coaches and mentors to younger years and girls aren't bullied if they aren't great at a particular sport - just encouraged and supported.
There are similar patters for academic excellence too.
It's almost as if it works to develop leadership skills, resilience and initiative.