As the maintainers of search, we want a coherent and cohesive code base to make maintenance easier in the long term.
Since the media search profile seems very cohesive with CirrusSearch, we should move that code into the CirrusSearch extension. We need to review that code more closely and see what (if anything) needs to be adapted to be more cohesive. In particular, we probably want to replace external API calls to ES queries (using new sister indices) and have a look at improving ranking (or at least ensure that the prerequisites are there to allow tuning of ranking).
Moving code to CirrusSearch should only be done after analysis.
 analysis of media search code
 list of things that need to be addressed created (as sub tasks to this ticket)
Links to some other MediaSearch related tickets:
T262522, T258054, T258053, T258052, T252692, T258055