An interesting little piece of info was discussed on "The Crow" Collector's Series DVD which just came out. On the commentary track, Jeff Most (the producer) talks at length about the soundtrack and mentions interest in having Curve and NIN for the two featured bands in the movie. However, somewhere in between talks and actual filming something couldn't come together which Jeff doesn't mention. Here's the related bit of info regarding Trent Reznor and NIN.
"We were looking for one female fronted band and one male fronted band. The male fronted band which appears in the film is Thrill Kill Kult, was originally committed to be Trent Reznor, and his band NIN, was actually the first person I ever spoke to or should say artists I approached to do a song for The Crow."
Source:
Slaw