Killing people with aids doesn't destroy the virus itself, it just minimises transmission. I'm not sure this is a very ethical method though, it might be better just to castrate/? (There's no equivalent for women...) everyone with aids, which would stop the main transmission route - I don't think approval for such action would be high though.
Yes, you are right, unpopular. Such a serious illness needs more than half stepping, after the killing off, people will have to swear off monkey sex as well. This is going to be tough for some.