After investigating via a support ticket, I learned this about the Topic Banner widget (the one that features a topic/event atop the community).
I've heard back from engineering that this particular widget is not intended to support the private visibility function, so the team will be removing those settings from the widget.
This is specific to the Private visibility setting, but (I believe) that’s core to the entire widget visibility settings (it’s private and then visible to only certain segments). So effectively the Topic Banner widget can only be used in Public mode and cannot be customized via widget visibility settings.
While disappointing, it makes logical sense. When visible and in-use, the Topic Banner displays across every single page of the CC platform. That’s an outlier from most other widgets, which are specific to customizable pages (Home, Community/KB/Ideas landing page(s), Custom Pages).