I like the automated aspect of this. I wonder how this could work with the group grading workflow that Damyon has been reviewing for Assignment. Seems like this is a perfect fit for a step in a workflow. Both tasks do, marking an assignment as submitted and then some post processing.
Completion won't remove the assignment from view, and I was thinking conditional release on the x number of assignments the student didn't post to. So you said when they submit to 3 assignments they don't have to submit anymore. So I was saying conditional release would hide the 3 assignments that the student didn't do.
One other thing you may want to think about is the the gradebook. I know in the past in this kind of scenario teachers run into an issue of each assignment having a record in the gradebook even though the student doesn't need to complete the assignment. Here it is most important for the student in the User report, or creating your own student facing grade report. I don't believe conditional release works with the user report to not display an activity a student can't see. i might be wrong on this I haven't played with conditional release extensively.