When a user is given the "Manage app" permissions, they are able to see all tabs, including the Agents tab, under Playlist setup. Playlist stores information about how tickets should be assigned for each agent on the User object in Zendesk. Since Zendesk only allows Admin users to modify Agents, only Admins are able to assign Playlist rules to agents.
Although not ideal, the only workaround is to ensure that the user is set to the Admin role if they need to assign Playlist rules to agents. Improving or providing more clarity around the "Manage app" permission is on our roadmap.