As you suspected, there is more than just car replacement taken care of by insurance (some of them are pointed out in Chad's storyChad's story:
- Rental car — usually an insurance claim will be accompanied with a rental car if the repair will take several days.
- Lost wages — it's possible that your coverage will include lost wages for time taken to manage an accident or for hospital visits or stays. Obviously, your personal health insurance may cover the bills but not the lost wages.
- Uninsured driver coverage — According to this source, it is required by some states but not most. Sure, you can still just replace the car with your fund if you need to, but the risk of damage to your car by an uninsured driver is higher than the risk of you wrecking it yourself.
- Comprehensive insurance — At my insurance company, you don't get premium increases for a limited number of comprehensive claims. I've made a couple - damage from a large rock smashing a hole in my bumper on the freeway, a spill on the cars interior. My return from this coverage has been far better than my cost.