There's a technical problem and a practical problem with this.

The technical problem is related to how to determine whether an assignment is published or not. With this change, and suppose we start an assignment for one student but not for the rest of the class, we have to treat the assignment as published in some cases (e.g. the structure of the assignment cannot be changed any more), but un-published in some other cases (e.g. other students cannot access the assignment). It's not too big a deal but it does complicate the code and hurt long-term maintainability.

The practical problem is that allowing late submission only affects one student, but allowing early start may affect the whole class if the student leaks the assignment.

Overall I'm leaning toward no for this.

rabbott wrote:

In all the cases I actually use the individual student takes the test after the rest of the class. Would that work?

In that case you can just extend the submission time because if the whole class already see the test, the student would probably know about it if he or she really wants to. Also "allowing individual start time but only if it's after the start time of the assignment"
 is actually worse than "allowing individual start time" in terms of handling special cases.