Thanks for sending me the link to the service. I'm not sure why you can't add it, as I am able to. Try completely clearing your browser cache and then attempt to add the service again from my content. The add attempt did fail the first time for me, but after that it worked fine. You might ask your server administrator to check the log files to see if any request are failing. Also, try it from another computer or using a different browser.
Just FYI, the URL you sent me is to a map service. You mentioned your desire to support editing and to do so, you need to host a feature service (there's an additional "Feature service" capability you need to check on when you publish the data to your server). Since you are able to add your service directly to the map, given a feature service, you'll be able to edit in the map as well. From an editing perspective, there is no difference between adding the service directly to the map, or adding it to My Content first.
When you add a service as an item to My Content, it means that service can be found via a search. It can also be added to a web map via a search, rather than having the URL. WHen you add a service directly to a web map via a URL, the service is only accessed in that map and is not available for a general search of ArcGIS Online. But, as I said, the service functions exactly the same inside the web map.
Thanks,
Mike