I have been volunteering at primary schools, so there are a few reasons that make me feel that the slippers are needed. I think historical reasons too have a part to play here.

Japanese buildings used to have wooden floors that can make noise with shoes that were used outside. The straw mats lines on the floors, too, can get easily damaged with hard shoes.

With school kids and dusty playgrounds, I think it is a good thing to have kids not wear shoes inside the classes. wearing only socks its not so good, either, because of the legs of desks and chairs. Indoor slippers offer just enough protection.

The kids use indoor shoes for the gym.

Japanese people are clean people. As you already know Japanese people wear slippers inside the house not only schools :)