The problem with the instructor deciding when a student can test, especially if its a large dojo with lots of students is that it would be really hard if not impossible for an instructor to keep track of every student's progress. Imagine if you were running a dojo, you have over 100 students, and you have to know by heart where every student's progress level is and in addition to that it keeps changing as students gain progress as they keep working at it. That's why students should choose if they want to sign up for an up coming promotion exam, A sensai could not keep track of everybody.
If you run a dojo and can't keep track of the progress of people you teach multiple times every week, especially if you're using a belt system and so you know more-or-less where they are by looking at their waist, then you need to be paying more attention to your students.