Historically we've treated the contents of the $search query as a total black box to mongod. Most of the time, we never unpack the query from BSON, and it resides in the 
InternalSearchMongotRemoteSpec.mongotQuery as a BSONObj blob until the query is dispatched to mongot.
 
The one exception has been the "returnStoredSource" argument, and now (as of SERVER-96059) the "sort" argument as well. We should consider making another IDL struct for the contents of the mongot query so it's easier to access those fields in mongod/s.