First, having a knife is not the same as having a weapon; a knife only becomes a weapon if you intend to use it as such. This is likely a good thing about the way the law is actually worded. While the police may sometimes use terms like "legitimate purpose" with respect to knives, the law simply says you can't carry one with the intention of using it as a weapon.
In the UK the law is worded differently. Penknives are presumed to be tools, but knives with blades that are long and/or fixed are presumed to be weapons. Both presumptions are rebuttable. In a case like the one you cite, Puddy couldn't just claim the 15cm blade wasn't carried as a weapon: he'd need to show what other purpose it was carried for.
In my view the UK law is a reasonably compromise, given that we want any kind of knife control at all (which I appreciate many Americans won't agree with). The biggest problem with the UK wording is that when the statute defined "penknife", it didn't allow for folding knives with locking blades. That the blade locks makes it a safer tool, but the courts have construed it as instead turning it into a fixed-blade weapon.
So I don't think that banning blades has to be all or nothing. I also have sympathy with the view that if you merely make it illegal to attack someone with a knife, then you can't get involved until it's too late and someone has already been stabbed.