Croquet is the perfect game for schools. (Assuming the mallets don’t become weapons).
Adults and children can compete as equals. It is a game of calm, rational, strategic filth. A game where unsportsman like conduct and generally stuffing up your opponents is actively encouraged.
The absolute joy of watching a child give their toes a firm rap with a large wooden mallet is absolutely free.
Set up a lunchtime league and I guarantee you that your most competitive boys will be all over it. There’s even a National Association.