The video I linked to upthread shows officials removing a tranq’d elephant’s tusks with a chainsaw, a couple of inches outside of the skin. Didn’t seem to be any problems after the fact.
Rhinoceros horn is used in traditional medicine, notably in Asia, but not as an aphrodisiac. That’s just a myth, though it seems to be quite widespread. I agree though that there are drugs for treating all the ailments rhino horn is actually prescribed for, and those drugs are both cheaper and more effective. Everywhere there are stupid people who believe what they believe and are skeptical of science.
You know, now that you say that, I think I’ve seen that too. Maybe the “living” part of the tooth is up in the gums. Huh.
Actually, right now, the big use seems to be to treat over-drinking. Since horn can make you vomit, and vomiting may be of some small value, it could actually work to a degree. Mind you, it’s marginal, and they are better things to make you vomit. Justin Beiber music videos for example. 
But right now, it’s mostly Vietnam.
But it has no real medical use. It does not actually cure anything.
Probably because the officials and gamekeepers are careful and the poachers aren’t, which again goes back to the issue of giving a shit about the survival of the animal. I don’t know that the poachers will stop at the skin when taking tusks either. Every inch is worth money, isn’t it?
I think the UL reasoning goes: aphrodisiac = penis = things that look like penis = rhino horn. Yeah, rhino horns are used in Chinese medicine, but it’s for non-sexual medical use (of questionable value, I don’t think I need to say). If a boner is what you want, there are other things they use, such as deer penis.
Rhino horn is also a traditional material used for personal daggers in some cultures in the Middle East, so it’s not just a medicinal thing.
An AK-47 is cheap and easy to get in Africa. That, and the lack of concern for the animals, is your answer.