Can't add music to iPhone anymore
There could be several reasons for this as far as I can tell, but my solution, after hours of research was rebuilding the iTunes library.

You might be able to check if this will actually work by addig some new music to your library and then trying to sync it to your device.

If becomes available (black instead of gray in the Music section), then this might work for you.

The steps for rebuilding your library are pretty simple, described here: https://support.apple.com/en-us/HT201561

Basically: 


Close iTunes
Go to your iTunes Music folder
Move the itunes library xml to a different location (e.g. desktop)
Delete the .itl file (the actual library information)
Reopen iTunes and import the moved xml (Import Playlist)