Not nessasarily. The 'additional weight' could still be something that requires a suit to take advantage of directly, but which the threat of compels behavior. For example, many localities have rental laws that compel the landlord to pay thrice what they would otherwise owe in certain circumstances. E.G. if my landlord simply refuses to refund my security deposit without proper reason, or without following proper procedures within 45 days, then I can sue for 3x the deposit and attorny fees.
A similar thing can be implemented for vehicular damages. Something along the lines of:
1) If you are found liable and
2) Were given adaquete notice and oppurtunity to pay and
3) Ought to have known you were liable
Then the defendent is entitled to punitive damages for your making them take you to court. You don't even have to go as far as most tenent laws go. You could allow for a loosing defendent to argue that there defense was reasonable, (or reasonable given evidence available to them at the time), and that they should therefore not owe punitive damages.
Of course, this still is only tangentially related to insurance.