Latest update - I have discovered the cause of my issue. The map service, which is hosted on ArcGIS server, has many layers which have been grouped into categories e.g. Boundaries, Assets, Topographic etc. I had set some of these group layers on by default because some of the sub-layers need to be on by default. The issue of ALL sub-layers being turned ON after using the Near Me widget occurs when those sub-layers are under a group layer that is also ON by default (even if those sub-layers are not configured as a search layer). Sub-layers that are under a group layer that is OFF by default remain OFF after using Near Me.
I re-organised my map in ArcMap so that any layer that needs to be ON by default sits under a group layer that is also on. All other layers that I want to remain OFF by default sit under group layers that are also OFF. This is the only way I can get the Near Me widget to work the way I want it to (layers to remain OFF after using it).
However, a separate issue has also been found which Esri has been able to replicate and will be reported as a bug. This issue is that the layer state is inconsistent after using the Near Me widget under certain circumstances.