My initial thought is that this would create a "wild west" type society where bounty hunters go around and have shootouts with criminals, but without police training. Also, how are cash fines going to work against broke criminals? "Oh, you stole a bunch of money because you have no money. Pay a fine"
As I understood it, the legal liability insurance pays. The criminal then has to deal with whatever repercussions are laid out in their contract with the insurance company.
And if the criminal refuses to submit to the judgement of the civil court, that's contempt of court which needs to somehow be enforceable.
You could make the contempt of civil court a criminal matter, but then the same insurer ends up having to pay restitution to itself and this doesn't seem to end well.
Or if the civil court has its own enforcement arm then we don't seem to have moved enforcement to the place he thinks we have.