You should be able to grab any old files from time machine by clicking the little clock icon in the toolbar and selecting "enter time machine". I find it hard to suggest any course of action, since I don't understand exactly how the problem occurred - and I won't have to live with the results like you will if you mess it up.
Personally, I would have cloned the disk before doing anything, as I suggested above. Of course it's too late for that now, but I would still make a clone of disk in its present state, so you can always at least get back to that. Carbon Copy Cloner will run for free for 30 days, so you could easily do that (if you have a spare external drive).
You might then try deleting your iTunes folder and copying your backup to the Music folder, then see if that fixes it (but be sure iTunes is not running when you try). Perhaps a safer approach would be to leave everything alone for starters, then connect your external disk with the backup, start iTunes and go to Preferences > Advanced and change the iTunes media folder location to the copy of that folder on your external disk. That should let you test the backup copy and see if it works correctly. Of course, it also has the potential to corrupt your backup copy - which is why you need a backup of the backup!
If that seems to fix things, you could then set this preference back to the default location on your internal disk, quit iTunes and then replace the internal folder with your backup copy. But again... proceed at your own risk, and I definitely wouldn't attempt any of this unless I had good backup copies of both the old hard drive and the new hard drive.
Really, this points to the importance of having a good backup strategy. Time machine is fine, but not a very good substitute for a clone. Time machine backups must be restored, which is a very slow process. It is much easier to restore from a clone, and you can also boot directly from a clone on an external disk.
[doublepost=1547571093][/doublepost]BTW, you could also try posting to this forum. There are usually some very knowledgeable people there, and they might suggest a different approach.
https://forums.macrumors.com/forums/apple-tv-and-home-theater.100/