Use new public API for getting proxy MediaRouter2 instance

The change is also pre-work for plumbing the UserHandle from the caller
and allowing cross-user Output Switcher.

Bug: 321969740
Test: adb shell device_config put media_solutions
com.android.settingslib.media.flags.enable_cross_user_routing_in_media_router2
and test manually.

Change-Id: I985e97c9e53e6977dfa103aefa1f61b55b9d7459