Back in the "good old days" you firstly had to be a Pilot (hopefully learn to fly), then a Sub Engineer (hopefully learn to fix it) before becoming a Supervisor (hopefully able to run a team but able to do any of the other jobs from Piloting to fixing)
More nowadays Supervisors are being chosen for supervisory skills or for ability to run team/deal with Client. This is fine as long as you have a team capable of flying and fixing! Not always the case. So purely being a people person will not make you a good Supervisor.
And of course there are the Supervisors who are made Supervisors simply because the company needs one – not because they can do the job.
My own opinion is that in an ideal world, yes the Supervisor is a team leader, overseeing others. Alas you also DO need to be Superman to be able to do everything when the rest of the team can’t or the Client is shouting and screaming while the rig waits on YOU 🙂