Recover playlists from iTunes iOS device backup iPod associated with an install of iTunes on a Windows PC; the Windows PC died, and the iTunes library was lost.  Most of the library has been recovered to a new install of iTunes on a new Windows PC by following the directions in this answer, but the process removed all playlists from the iPod.

I'm assuming the playlists were included in the backup made in the first step of those directions, but I can't find any tool for extracting them.  I can find plenty of programs that can copy playlists from an iPod (might have been useful at the beginning, but not now), and plenty of programs that can extract anything but a playlist from an iTunes device backup, but I have not been able to find a program that can recover playlists from an iTunes device backup.

How can I recover the playlists from an iTunes backup of an iOS device on Windows?
I'm fairly sure that the playlists are considered data synced from iTunes, and therefore aren't included in a backup.

https://support.apple.com/en-us/HT204136

I just took a look through a few backups just to make sure, and I didn't see the playlist data anywhere.