Banning peanut butter would help with that, but it couldn't stop there. Plenty of packaged foods contain peanuts or other nuts, or are processed on equipment which processes those things. Foods are often repackaged by parents such that labels are not available.

My understanding of this type of reaction is that it doesn't require a large volume of the allergen to cause a very serious response. Trace amounts can be very dangerous. So it seems impossible to guarantee prevention by banning foods.

In my opinion, kids need to be aware of what they can and cannot eat, and the school must be prepared to actually deal with a reaction situation instead of just saying 'look, we banned peanut butter, what else do you expect?'