Why does the Jailbreak need to be untethered? Is the teacher walking around and making every student reboot in front of them? Rebooting takes a solid minute, so in a not-particularly-large class of 20 kids, you’re dedicating a solid 20 minutes to this activity.

Once Jailbroken, I don’t think you’d need a fake app. Just make a very simple patch that overrides whatever is preventing you from returning to the homescreen, and set up the patch to be enabled by a secret Activator action, like pressing the volume buttons in a secret sequence.