Hello Ethan,
I have been there too, regarding users not following policies, so I definitely second Floyd's advice. Remove backup capability from teachers:
https://docs.moodle.org/27/en/Course_backup#Course_backup_capabilities
and enable the automated backup module:
https://docs.moodle.org/27/en/Automated_course_backup
Anyway, you can manage (download, delete) the backup files from the restore page:
https://docs.moodle.org/27/en/Course_restore#Restoring_a_course_backup