[QSDetailedView] Fix Cast details view always showing empty view

The Cast details view, which uses MediaRouteChooserContentManager, was not
correctly initialized with the required `routeTypes`. As a result, the
manager would not search for any Cast devices, leading to the view
always displaying the "Searching for devices..." empty state message.

It's fixed by setting the `routeTypes` on the newly created
`MediaRouteChooserContentManager`in the `CastDetailsViewModel`.
This ensures that the manager is configured with the correct media route
types before it starts discovery.

This aligns the initialization logic with how the older MediaRouteChooserDialog
is typically configured.

Bug: 378513588
Flag: com.android.systemui.qs_tile_detailed_view
Test: Manually verified. Existing unit tests still pass.
Change-Id: Ia41c61e70e7e7b0d4b1e62ed27e1a82f315bb05e