This is the same logic that failed in the "War on Drugs". First off, there's a false dichotomy between proscribing a behavior and encouraging it. And strict enforcement of a "crime" that it turns out lots of people commit on a regular basis doesn't really work so well.
Off the top of my head: Many of the drugs in question are responsible for far fewer deaths than cars are, and so it's not obvious why the penalty fits the crime and there's a strong argument for outright legalization. Many of the drugs are physically addictive and not merely something people want to do with full free will; many people arrested would be happy to stop doing drugs if they got treatment instead. The war involved foreign policy and foreign military activity, whereas nobody is suggesting that we bomb Toyota if they refuse to make slower cars. The war has been waged particularly aggressively among historically-oppressed races and classes (cf. The New Jim Crow), whereas no directly analogous danger exists from enforcing low speed limits.
Just compare the streets of Tokyo to that of San Francisco and tell me it is not working!
Also, compare driver (and pedestrian) behaviour in Japan. In urban areas, both largely follows the rules and it is safer.
(Strict liability for drivers may also be a part of it...)