I think half the problem with crossbows is that they're easy to shoot accurately - like an air rifle - and they have novelty value. They're also easily concealed and don't need setting up. Contrast this with a recurve bow that's the thick end of 6 feet long, takes 10 minutes to set up and a considerable amount of training and practice to hit a target with any proficiency.
Plus, have you ever tried using a recurve bow for a drive-by shooting? No?
I wouldn't object to needing a licence of some sort if I thought it would make any difference. But it won't; I don't think I've ever heard of a bow being misused or used in a crime. Perhaps it's time to restrict crossbows though.