"Visibility above cut plane" should be customizable from the "Visibility/Graphics Overrides". A drop down menu per category could have the following options "Default", "No" or "By Family". This also requires a new family parameter.
The Generic Models category creates problems in architectural plans, as this category is always displayed, even if it is above the cut plane. For instance, if a ceiling mounted element (or a wall mounted sun screen above the but plane) is modeled as a Generic Models, it will display on a architectural floor plan (where it should not). This problem is linked to the lack of categories that can be displayed as Cut, wich leads to use of the Generic Models category also when there is a more fitting category available. See the related issue Cuttable vs Non-cuttable Families.
Alternatively, we need a new default category that does not display in floor plans if it is above the cut plane (unlike the Generic Models category). Better yet, make the Generic Models category not display when it is above the cut plane, and instead introduce a new category "Generic Models Overhead" that displays even if it is above the cut plane. This would at least be more intuitive than today.
Todays behavior is documented here:
@Anonymous, @Thierry.A, @post8 and @kh_jea voted for a similar idea that was archived a few years back:
https://forums.autodesk.com/t5/revit-ideas/solving-generic-model-families-problem-with-view-range-and-plan/idi-p/6473760
@james.levieux mentioned something similar in a comment in the related issue "Cuttable vs Non-cuttable Families":
Cuttable vs Non-cuttable Families - Page 2 - Autodesk Community