My suggestion is to turn it on in the battery saver as well and not just on the "background apps". In settings, search for battery saver and add Groove to the list of apps that run on battery saver. This seems to be a funny solution, but my Groove music is playing a little better in the background with this setting on.
Syncing from the desktop is by two ways. And both are kind of buggy even now.
- create playlists in Windows media player in Windows 10 and sync it to your phone. To do this, connect your phone to the Windows 10 laptop, open Windows Media Player(Not Groove) and in the libraries, you can sync with the phone you connected. I use this only to sync (kind of)short term playlists that will have like 10 or 15 songs only. I use this to listen to new albums that are coming in, listen to them for a few days and if I do not like the album, I simply remove them in phone as well as desktop.
- Kind of more buggy way is to use Groove Music Desktop app to Create a Playlist, add songs to it. And when your phone is connected to the system, it will sync automatically. This is very reliable if you add a few hundreds of songs at a time and let them sync with each other. Not recommended for thousands of songs at a time. If you are happy to do the exercise again and again, like 1000 songs per connection, this is more reliable(in terms of correct album art and other media properties). I used this at the beginning of syncing songs to my mobile. Never had to use this again. With all the improvements happened, I think it will be safe for you to try 5k songs at a time.
I know these are not wonderful solutions for a huge number of songs. But if you have patience and are willing to split the job and do it over and over again, these two are more reliable than Groove Music(mobile) automatically syncing things up for you. If we have the wireless sync(Zune and my old lovely Lumia 900!!), the above steps will be better. But this feature might only make a come back by this year(??!!). So until then, USB FTW!!!