Instead of the separate set of EME specific Categories, I would like to be able to add an Event to an existing regular WordPress category, so that I can pull up its content in category listing pages and category widgets.

Perhaps this is not possible unless events become posts; I don’t know. (It works very well in my old version of The Events Calendar, in which events are posts.)

[ In any case I see that the EME Events List widget includes EME category support, so my sidebars should be okay once I redo them. ]