I also have a Zune HD. It is now relegated to mp3 files I own. Not even podcasts sync (well, not easily). It still works and still sounds great, so I'll still use it sometimes.
As I understand the basic problem, the DRM Zune used was the same as PlaysForSure. When Xbox Music came out, they got a new DRM which Zune could not use and Zune Software could not sync. It appears the player itself has to retrieve whatever DRM stuff it needs from the server independently of the download (in Zune DRM the rights came over with the files---no longer the case). Until very recently MS maintained two libraries of new and old DRM music. Since the marketplace was eliminated (at least access to it---it may still be out there somewhere) we can no longer download stuff the Zune can play.
I believe they may need new Zune firmware and new Zune sync software for the Zune device to be usable anymore (if even possible) and neither of those two things are likely due to the really low numbers out here.
FWIW, iTunes seems to have a similar issue with their DRM. An iPod does not work with Apple Music (the subscription service). You need a Touch or something with internet connectivity (yeah, Zune has it but somehow that does not appear enough).
The easiest solution is to use a spare phone as an mp3 player. That's what I'm doing. One advantage of this is Bluetooth which Zune did not have. You can get an eBay phone (even Android, but I've found Groove on that is not as good as a Windows Phone). I tried to sync my phone with the PC back with Xbox Music and was one of those frustrated that nothing I copied over would play. You had to leave the phone connected to the internet for several minutes while it downloaded the music rights. It works the same today. You can copy over non-DRM stuff, though, which works fine.
The whole concept of music storage has changed to a "cloud" thing. Your radio stations, playlists, collection all sync via the cloud to all your PCs, phones, etc.. Although I don't seem to have found a way for Groove playlists to sync across devices, which would be a nice feature since working on that kind of stuff on a computer is a lot easier than on a phone.