Despite the frequency of this issue, there have been few solutions, however that doesn't mean that this is an impossible situation to overcome. From my experience, there is a simple fix and it starts with getting everyone "on-the-floor" to not see themselves as different, at least in the ultimate reason they are there: to help save lives and provide the best possible service to the public and field responders.
This approach starts with a redefinition of the roles inside the center. Rather than seeing Supervision as only the responsibility of one single person who has passed a test, been appointed, or been selected, the roles of Supervision and Leadership should be seen as everyone's responsibility. From their earliest days on the job, new Dispatchers should be trained on effective supervision, why supervision is important, and what she or he can do to help run the floor in a more effective manner.
For example, if new personnel are trained (and assessed) on their performance in: Admin Phone; 9-1-1 Phone; Radio; EMS Dispatch; Fire Dispatch; Law Dispatch; Support Dispatch; and Basic Supervision/Leadership; then they have a much better understanding of the roles played by each member of the team. If policies are written to encourage team performance and, to a limited extent, team accountability, then everyone has organizational support to see themselves not as an island, but as part of an overall group of professionals.
The next logical step would be to have Dispatchers on a shift rotate a "lead Dispatcher" type role that may serve as an Assistant Supervisor. (Likely after the completion of their probationary period). In this way, every member of the shift would take turns seeing things from a supervisory prospective, with responsibility for problem solving, crisis resolution, and related tasks. For many reasons, the traditional supervisor may need to be retained, but the ultimate goal would be for the person in that role to be able to serve in a largely oversight capacity, developing each member of the team to have skills at all roles on the floor, including supervision.
In case you are wondering about whether or not this would work, I can tell you from my own experience that it does. In FDNY, dispatchers served in the "lead" position (known as Decision Dispatcher) as soon as six months after completing initial training and, with effective mentoring and supervision, this helped develop exceptionally skilled personnel.
Maybe it will not work for everyone and it certainly requires a different approach and mindset, but it can pay significant dividends to bridge the divide between the different roles on the floor and encourage everyone to help solve the supervision challenge.