As a person who has read more than their share of these types of essays, I think that the challenge is that many students do not know how to write this type of essay well without it reading like they are making excuses for their performance. Very few of them are actually very well written and all read the same and can be pretty interchangeable.
I believe that because it is going to be seen on the transcript and if the student has grown, it is something that the counselor. The counselor has more opportunity to take as many words as needed and even give a few anecdotals from her peers regarding his performance.
Perhaps one of his recommenders, who knows the student well and his struggles could address his progress and growth.
Student should use his essay to highlight something new about himself that is not readily on the application.