And the Borg are not really a villain. They're a race; a set piece. And they're primarily known for their association with the next generation. It seems odd to abandon the next generation crew, but then to use their same trappings.
You could argue that they were not a villain originally, and were more like a force of nature, but once they introduced the Queen with her own independent thoughts, emotions, and shortcomings, they did become a villain, or at least were led by one. Hopefully they'll go back to the original force of nature concept if they do decide to reuse them.
While it's true that the Borg are most associated with TNG, VOY comes a close second, with ENT and DS9 bringing up the rear with all of them having Borg appearances. So only the TOS series/TOS-era films & film segments and nuTrek have not had a Borg appearance yet, and even then, there are indirect connections and tie-in materials which have featured the Borg (El Aurian refugees fleeing the Borg on the Ent-B, V'Ger if you go by Roddenberry's hint about the planet of living machines, the Borg technology on the Narada, etc.).
I was burned out with all the Borg appearances before and the way they were defanged, but now that the TNG-era films and series are probably dead, I wouldn't mind at all seeing them get a reboot of their own, especially if they can become the faceless unstoppable menace they were originally.