There are a myriad of ways to approach this and it really seems to depend on what works best within your leadership and ministry organization and with the personalities involved. For example, is the tech group seen as being a ministry or is it solely a functional group with ministry served through other groups? That could definitely affect what works best for you. So could things like the relationship to a Worship Leader or Minister of Music or whatever other roles and organization already exist.
Just as an example of how this can vary, some churches have everything related to the audio during services be under the Minister of Music. Where the MoM has a good relationship with the head of the production/tech staff and works with them that can work quite well. If the MoM tends to dictate and use the tech staff as the scapegoat for any problems, it doesn't work well.
Maybe I misunderstood but it seems like you are saying that the
Production Manager or Production Director or whatever is quiet during services. Is there someone else calling cues and handling the overall running of the service? It just seems odd that the person that coordinates all the tech aspects leading up to the service then has no role during the service. A simple instance would be going from the pastor speaking to a video, who handles calling the
cue so that what's on any screens or displays, the proper lighting scene, the related audio source selection and mic mutes,
etc. all occur in sync and at the right time?
I find the titles you used interesting. I typically envision "designer" as addressing someone involved in the creative aspect and "operator" to the actual operation. In many cases the roles may be combined but some churches may have a designer and several related operators. You might also want to think about the "Audio Engineer" designation. While some states legally limit the "Engineer" title or practice of Engineering to properly licensed Professional Engineers, even if that is not an issue using the "Engineer" title for someone with no Engineering background or license can be less than well received by the licensed Engineers in your congregation. If your audio person feels the Engineer title is so important that they won't accept being a Technician or Operator or whatever else like the other members of the tech staff then maybe there are some other issues to
address.