Cannot hide visibility/private field

The "Visibility" field (`"field_name": "private"`) is always shown. Trying to hide it or change it results in an duplicated field.

Workaround: override `organization.html`. Should those two field be handled independently?

Start by examining the organization.html override mentioned in the issue and trace how the Visibility field with field_name "private" is rendered. Reproduce hiding or changing that field, then verify that visibility and private-field handling can be configured independently without creating a duplicate field.

Written by the indexing model from the issue text.