The Borg's crossover appeal also comes from the fact that they are such an overwhelming and unexpected enemy that no matter who meets them for the first time, it is going to be an extraordinary deadly adventure that's going to make some good cinema.
It is harder to mess up. Even ENT: Regeneration was exciting to watch. And having 21st century humans like Lily meet them in First Contact was, in my mind, even better than the Enterprise D meeting them in Q Who. I don't think they are going to pass on the chance to have Star Trek's first crew have their own first encounter with the Borg, which is going to be something. I don't think you can go wrong with it.
Voyager messed it up because we and the crew got to know them, plus they approached the matter in the wrong way and rushed through the suspense. Which is not to say I didn't love when Janeway met Seven in ENT: Relativity - that's another thing I find appealing when it comes to firsts, and another reason I want Voyager rebooted and done right (call me weird).
But the formidable, relentless enemy you just meet that defies any expectations is far from overused, it hasn't been used enough, not even on the small screen. To fuck it up now, you'd have to give them an utterly dismal treatment – which given the first two films ain't gonna happen. Which is another thing that Voyager missed doing – encounter worlds just facing the Borg and get assimilated, and be powerless to do anything – like VOY: Dreadnought without the happy ending.
If you think the Borg are a bad idea presently, just wait until future film makers decide to do the time travel stories Borg in WW2, Edith Keeler is Borg, Borg in the Wild West, Borg and the American Revolution, Borg and the Spanish Inquisition, Borg and the Romans, Borg on Apollo 13. I fully expect that those will happen before I die, and I also fully expect that some of them will end up being my guilty pleasures.
ETA: I do prefer an original story without an old enemy. But the Borg – which are very likely - will work splendidly.