Closing for disinfection works in part because people don't go to those places while it is being disinfected and thus don't spread things in them.

It’s more complicated than that, and closing for four days is somewhat like a reverse four-day quarantine. You have twenty kids starting to show symptoms that get their slobber all over everything (you have no idea how gross K-12 students can be, btw). They’ll stay home when the symptoms get worse (parents aren’t hell-bent on sending them to school, but would prefer to do so and as such will be in denial until it’s too obvious), but now they’ve already gotten all the high-contact surfaces contaminated: keyboards, mice, desks, toilets, chairs, water fountains, door handles, telephones, etc.

Kids don’t wash their hands. Kids don’t have a concept of personal space. Kids don’t cover their sneezes or coughs. Kids share food, containers, utensils, etc. such that even if you eliminate direct aerosolized contamination, the entire school is a germfest. One kid with a weak immune system picks it up from contact with an exposed surface, and the whole thing starts all over again.