Do supervisors need access
In a perfect world, supervisors would access appointments. They would directly go there to submit appointment requests on behalf of themselves. They could also review their existing subordinates, check on the status of an appointment request, etc.
Supervisors (unless otherwise privileged) would not have access to the inbox or any other page besides the my employees page and the corresponding employee detail screens for their subordinates.
They do not need to have any of the defined roles to gain access. To submit appointment requests, they must have been granted the ability to submit specific appointment types by a system manager. This is done on the authorization page in the system.