CastDetailsView: Migrate functionalities to chooser content manager

Move all the necessary functionalities from MediaRouteChooserDialog to
MediaRouteChooserContentManager.

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

Bug: 395947047
Flag: NONE refactor
Test: http://ag/32396457, atest MediaRouteChooserContentManagerTest
Test: Cast to a device, click on the Cast tile, launched the dialog and
verified that the chooser UI works.

Change-Id: I423379bfdaf38fece2243de2cea29daeb2f6df79