CastDetailsView: Remove route object in MediaRouteControllerDialog

Remove mRoute and mRouter in MediaRouteControllerDialog, since these
variables should be in the content manager only.

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: I8f43c98fef504004cedda688c3d691788da948e0