To clarify, iTunes has its own music library. But you need to copy the files you downloaded to that library. There's various techniques to do this properly without side issues. Follow 6_pac's link to see how.

As 6_pac pointed out, the LW playlist you see in iTunes is simply a playlist but the files themselves are not within the iTunes music library. That is something you need to fix by adding the songs properly. The LW playlist in iTunes is highly misleading. I hope you see the logic now after seeing 6_pac's link.

PS: I'd recommend disabling the LW playlist in iTunes. And simply add the songs to iTunes when you can. In that way you won't be mislead into thinking the songs are already in iTunes music library.