Sure a sharpened edge is low tech, however, there are some even more basic options that don't require much tech.
Someone who was serious about killing someone would in fact consider options other than a sword.
A sword is too cumbersome for most people, rather hard to conceal on ones body, a bit expensive to be used disposably, and is not quite the 'ultimate' weapon it is perceived to be.
According to a copper I've spoken to the commonest impaling device is actually a screwdriver. It doesn't require much in the way of sharpening. Just having a wedge and a stiff shaft is all that's needed to penetrate. A fine edge is more essential for cutting blades like for example a kitchen knife.
In terms of effective melee weapons one can come out quite well with stiff shaft like perhaps a walking stick or a pvc pipe.
So yes I agree the intent of the legislation is an over reaction to the perceived threat of swords being on the streets.
Following the concern of the legislation they might have to ban walking sticks from all those geriatric gangsters and watch out for those gangs of home-improvement hooligans...:-)