Why? It's all Star Trek.
It just feels fanboyish, and it feels like it would take more convoluted explanations to make it work. "Because Nero," won't cut it.
You spoke of a second best known adversary, but in reality, the first best known adversary (not Khan) is the Klingons. STID sets up the Klingons and they seem a more fitting piece. Granted, we've seen a lot more of the Klingons in the movies, it just has the better potential to fit this crew without seeming fanboyish. They can actually have a good villain without gutting the whole concept of the race. And they can probably make a more poignant piece about a war or the prospect of one.
It just feels out of place. The analogy I've used before is, let's say for some awful reason they decided to reboot Cheers. Then for some reason they decided to focus a good portion of a story in Seattle with Frasier's family and radio show. That's what it feels like to do the Borg.