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.