Do you think it always must be ‘lonely at the top’ for supervisors or that if it is not, you are doing something wrong?
You are asking whether a person in a supervisory position must always be lonely, assuming they are doing a good job. I would argue that this is at least generally true in the context where the person is a supervisor. In other words, they will need to be somewhat lonely in that capacity, but they need not be lonely in the rest of their life.
Supervisors can include bosses at work, teachers, coaches, or even parents. They are people who are entrusted with making sure that others do what they are supposed to do. This is why supervision can be a lonely thing. Supervisors must in some ways maintain an emotional distance from those they supervise (of course, this is not true of parents). They cannot get too close to those whom they might have to discipline, to fire, or to fail in a class. For this reason, they need to remain a little detached.
This is not to say that supervisors cannot or should not be friendly with those they supervise. It is important to be friendly. But at some level, they have to maintain a distance. They can not be on truly intimate terms because they must be able to do things to the supervised people that might be unpleasant.