While I like the general idea of such an "insurance", I think you may forget
that some laws are put in place to PREVENT people getting injured, such as most
rules in traffic:
2. Those who are accused of traffic violations for behavior that did not harm anyone
And of course this is not only true for traffic alone. It's not only about
punishing people that harm others!
Would you drive by a school with 80 miles/h?
Would you allow people to carry a gun in an air plane?
Would you allow lose regulation in banking and finance so that intelligent people can easily rob not so intelligent people?
Haha there we are right in the middle of the mainstream vs austrian debate.
The austrian perspective is that people would be(come) self responsible (and drive by schools slowly) in order to prevent being sued in case they actually harm anyone. The rationale is: If there is no big brother you have to justify your actions towards you will justify them towards yourself and towards others (or towards overall consciousness, see
https://bitsharestalk.org/index.php/topic,20582.msg265556.html#msg265556).
It also relates to the subjectivity / objectivity debate. Austrian economics says there is no objectivity. So a state can not possibly set sufficent and objective limitations on people's behaviour. A market mechanism has to do it. In this case a quasi market mechanism would be that people judge their own risk (being sued) / reward (time saved by driving fast) ratio.
With banking the austrian perspective imo make a ton of sense: I would say that the reason there is so much "risk" in the system and the reason that there is no financial education is that the state is absorbing all the risk. But that just increases the overall risk that (big) financial players take which get all their play money from the little man that has no risk awareness. If there would be no regulation in finance whatsoever the little man would certainly not give his money to a bank which is taking big risks.
There are other areas where an application of pure austrian economics is at least difficult which is global externalities such as CO2 production or other outputs of the production / consumption process. Here the austrian perspective would practically suggest that farmers in Bangladesh collectively sue those that were the big CO2 producers over the last centuries.