Your argument is fallacious, because the law covers much more than death threats. It makes illegal to send messages “for the purpose of causing annoyance, inconvenience or needless anxiety to another”, not just to send death threats.
It has been illegal to say things in public, or to send written material through the postal system, or to use telephones “for the purpose of causing annoyance, inconvenience or needless anxiety to another” for very many years. This new law just ties up loose ends with online activity.
