Here is my take on this. In a high school setting, where the goal is education and
safety I can understand a staff TD supervising in the booth. This allows the TD to help the SM, and board ops if needed, and supervise the performance in general. However, student TDs, unless they are running the show, probably shouldn't be in the booth. Why? Because, I hate to say it, but most high school tech people, especially upperclassmen, suffer from "big fish in a small pond" syndrome. A student TD is more likely to comment, try to advise, or even just jump in because they think the op is doing something wrong. This would be detrimental to the op learning something. It just comes from the fact that most (not all) 15-18 year-olds just don't have the experience to be effective teachers.
In a college setting, I think that you would find it very rare that the TD would be in the booth for a show (unless the TD is running the show). Usually, in the college setting, the TD's job has nothing to do with the programming or running of lights, sound, or
stage management. Also, as with many positions, the TDs job is usually done at opening night unless scenery or
props need to be repaired or replaced.
In the pro world, it is usually pretty similar to college. Unless the TD is running the show, you won't find them near the booth. One would hope that in the professional world you are hiring ops and SMs that know what they are doing and don't need to be supervised.
Also, not every
theatre is the same, not every school is the same. So, what I have said above is a generalization, and not done everywhere. Also not that my use of op refers to operators as opposed to "original poster."