This is getting better but it's still needs some refinement. The Badge widget shouldn't require a controller, just a child. A factory like Badge.count(), that created a private Badge subclass might handle the special case where the Badge's child displays an integer with a limited number of digits.

Updating Badges based on application state, by rebuilding the badges themselves or some part of the navigation bar or its ancestors, should be up to the application. An example that demonstrated a simple version of this would be helpful.

Since the typical use-case for a Badge is displaying a short string within a filled "stadium" shape, it makes sense for it to have shape, backgroundColor and textStyle parameters. The TextStyle would be used to wrap the child in a DefaultTextStyle. A TextStyle parameter is a little tricky because the common case is to just override the default TextStyle's color. If you also provide a textColor parameter, then it should probably override the TextStyle's color, even if that's non-null.

This widget will need to deal with textScaleFactor and (probably) visualDensity to the extent that it constrains its layout to fit into a navigation bar.

Badges might be applied to any widget, not just to a NavigationDestination. This widget would be more useful if it had a child and an alignment property. That way it could be used to decorate NavigationDestinations as well as any other widget.