Why not?
If someone is too dangerous to be on the streets (but only if they can somehow get their hands on a gun, which they definitely can), then they shouldn't be on the streets. The only other option is that they're not too dangerous, even if they own a gun, in which case they should be allowed to own a gun.
of course there are dangerous people with no criminal history, but no background check will catch them, so nobody can.
If I sell a car to someone, I don't wonder how many dui's or speeding tickets they have. It's not my job to decide if they're too dangerous to own it.