It would be nice if layer files shared in ArcGIS Online would be accessible in ArcMap With ArcMap at 10.3.1 you can't add layer file (.lyr) from the "Add Data From ArcGIS Online" dialog or the Search window. Only map services, feature services web maps, and layer packages show up.
The people that I work with have lots of layer files that they want to share within their organization. They don't want to share these as independent web services in ArcGIS Online. They use lots of definition queries and different symbology in these layer files. So one dataset in the database might have many layer files. They want their ArcMap users to be able to easily access these layer files from within ArcMap. They would also like to share all this in ArcGIS Online instead of a search service in ArcGIS Server.
ArcGIS Pro currently has the ability to access layer files from ArcGIS Online. We just need the same support in ArcMap.