Allow the teacher to control who receives a particular "Submission Reminder" notification for each notification - such as students, family members, and include Homeroom Advisor as another option.

Also, somewhere, allow the teacher to track how often these notifications have been sent, while you can see what has been send in the Activity log for a particular student on a particular assignment, it is a lot of clicks to see that and you'd have to click through all the students to remember who you sent them to.