why isn't "Force Log out" a good solution for you? You don't want other students to be able to see who used the device?
It shouldn't be hard to change the app's code to delete the site automatically on logout, but the result would be the same as deleting it manually: the user would have to enter the site address too.
If you don't want them to enter the site address every time, the only solution is to have a custom app for your institution, that way the users will only be able to connect to your site and the URL would be fixed.
We provide a service where we can customise the app for anyone who needs it including the Push notifications (Airnotifier) service management. This service helps fund the improvement of the app in general: https://moodle.com/branded-app/