It seems to me that the software behaviour you are observing is by design.
I recommend reviewing Difference between map layer and spatial dataset (e.g. feature class, shapefile)? and then think of the:
- Table of Contents window as being the place where you do things with symbology
- Catalog window as being the place where you do things with data
Cartographic representations are about symbolizing and so you would expect to see more access to the functionality that involves them in the Table of Contents than in the Catalog window.
I suspect that the reason Representations can appear in the Properties of a feature class in the Catalog window is that they are stored as columns in a geodatabase.
The design appears to be a sensible compromise between having the data properties of representations accessible in the Catalog window without having all functions from the Table of Contents related to creating representations duplicated there.