The pre-Christian European unicorn was fierce, untameable, goatlike, white, beautiful, and REALLY male (pretty much a penis with hooves, an objectification of rape). But it also had healing abilities. The original virgin-as-bait mechanism was a challenge between the girl's purity and the unicorn's lust, and if the girl lost, she climbed on the unicorn's back and was never seen again.
So what happened? Visually, the unicorn didn't lose its beard, cloven hooves, and tufted tail until the second half of the 20th century, but the transmogrification from galloping penis to fluffy bunny seems to go MUCH farther back than that.
The blame probably lies with the medieval Christian church, which REALLY hated ambiguity and shades of gray. While the ancient Greeks had no real conceptual problem with the idea of a healer god who committed the occasional rape, the contrast would have given a medieval clergyman fits, so the beautiful white healing creature had to ALSO be an embodiment of purity. (The medieval church had a vested interest in the idea that you could be holy and still indulge in wholesale slaughter, so the ferocity got to stay).
In the interest of complicating things, I will offer the following bit of speculation: Maybe the unicorn gets its healing abilities from CONSUMING purity (which Dementia points out makes the gray REALLY dark...)