Wouldn't it be more analogous to if the state or the Department of Transportation paid for all accidents since they are the ones controlling the system? Insurance companies don't have any power to change road designs or traffic laws.
Also, "No Fault" doesn't cover vehicle damage. Apparently, this is what I was remembering. It changed when I lived in Pennsylvania (that was quite a while ago).
Several US states HAD No-Fault insurance laws--ie your own insurance company pays for your own accidents.
The insurance companies lobbied like crazy to get those removed. They're all gone now.
If you got those No-Fault insurance laws back, the required systems thinking would ensue.