Great question! There currently is not a way to hide the to-do list from students and activity stream. I think what may help prevent this is if you incorporate using gating on activities which will require they be completed before a student may move on in the course or complete those in the to-list before other activities. See How do I make an activity's visibility dependent on competency mastery? for additional information on gating.
Is there a reason why an assignment wouldn't show up in students' To Do list? They can see it in Activities and the settings look right, but it's not showing up in their To Do list and we are wondering why? Asking for a teacher...
There can be some reasons for this. One way is if the activity is not gradable. Another is if the number of days for the due soon setting in the domain settings is not within the range of the activity's current due date.