Extract and simplify RoutingController book-keeping operations

This change deduplicates code and extracts some methods as pre-work for
proxy routers keeping a local copy of non-system routing controllers.

This is a non-functional change.

Bug: 330300516
Test: atest CtsMediaBetterTogetherTestCases
Change-Id: Idcdf68e56913059683626575ed9e189e7b871ebe