We use Autonomous Single App mode on iPads for searching the catalogue and it works fine for that purpose but it doesn't help in this situation as we have many different games and apps available for the kids to choose from. I think that they're using the Control Center to turn off wifi, not the Settings app.
We are using Wifi Whitelisting so that the iPads stay on our network where they are authenticated by MAC rather than jumping onto other nearby networks where they would be caught in splash pages or otherwise prevented from "phoning home". There is no cellular on these devices. They are only used in the branch.