I had this problem in 2014 with my iPhone 6. I couldn't figure out why my songs weren't downloading to my phone. I would plug it in, iTunes would jump quickly past the "sync music" section of the sync, and then say finished. My songs would show up on my phone but were greyed out with stop buttons next to them, just like above. I swore it was a problem with iTunes Match, but I had never signed up for that service. It wasn't until I was about to head out the door to the Apple store that I figured out the fix.
My iPhone continued to sync even though it said it had finished. A small sync icon was moving on my phone and in the sidebar of my iTunes, but the status bar thingy at the top of iTunes said my phone had stopped syncing even though it hadn't.. so weird. It must be the result of some iTunes update. So DO NOT UNPLUG YOUR IPHONE UNTIL IT IS DONE SYNCING. I continued to check my iPhone and the songs slowly started to download to my phone. This was the same case with music videos, tv shows, and movies -- it just took longer to finish syncing.
So iTunes must recognize all the music it wants to sync, puts the names on the iPhone, finishes the formal sync on iTunes, and then downloads all the content afterward. I was pulling the plug too early and as a result I thought it had finished syncing but it hadn't. It isn't some glitch, it's just syncing! Don't be fooled!