CastDetailsView: Move callback and icon to controller content manager

Move the functionality of creating the icon and the callback from
MediaRouteControllerDialog to MediaRouteControllerContentManager.

This CL follows the refactor mentioned in go/cast-details-view.

Bug: b/395947047
Flag: NONE refactor
Test: Cast to a device, click on the Cast tile, launched the dialog and
verified that the controller works.

Change-Id: I42b7f334d51962ecd2c201bd44cd0e2fc634a72f