Grade school sports?

Would kids be better off if we eliminated all formal sports teams in schools K-12 and replaced them with sports programs?

The programs would consist of every field, court, track, etc. in all towns open to all kids all the time.  There would be hours when volunteers would be on site to monitor.  Basically the kids would show up and play, all day if they want.  Teams would be pick-up.  The monitors would be there to be sure all get to play equally.  For example, if a kid loves soccer, she/he could stay all day to play. 

All the money for coaches, AD's could be saved as the jobs wouldn't exist.  And all kids could play all they want at what they want - without the outside interference of parents/coaches/etc.