How I'm thinking about this, that makes no sense.
What is the use case that it would work this way?
User accounts MUST have a primary department. You can't save the agent account without one assigned. If I assign the default support department, I receive all notifications of all new tickets that I don't need to know about. So I created an ADMIN department for those level of tickets for me to manage.
But I don't want the public to create tickets in the web UI directly in ADMIN as most users have no idea what issues should require ADMIN attention. So I set the department to be private (internal use only), which makes it disappear from being an agent's primary.
IMO, this needs to be changed so that this works.