Correct. The state pursues a criminal case, and doing so is at their option. Certainly prosecutors give great weight to the desires of victims (thus why police will ask if you want to press charges), but they have no requirement to do so. This works both ways - the state can file charges against someone even if a victim does not want the case pursued.
The venue for a person to pursue another person for wrongdoing is civil court, not criminal court. Ultimately, because insurance was involved here, the best bet is to have the insurance company handle this piece. If the other party does not have insurance (or said insurance does not want to settle), the company may chose to pursue a civil claim against that person (which would be covered by insurance if the other party had it). As an example, I once lived in a condo building that experience a fire due to an improperly disposed of cigarette. While I thankfully had no damage to my unit, the city closed the building until the damaged areas were repaired and the entire building could be brought up to the current fire code. My homeowners insurance paid over $16k for me to live in a hotel for several months. They ultimately retained a law firm to pursue compensation from the offender in civil court, but alas she didn't have much in the way of assets (she was renting the unit and had no renters insurance) and they ended up dropping the matter. Thankfully my insurance rates never rose from the incident and I've remained loyal to that company ever since.
Theoretically, the OP could have pursued the other driver for damages in small claims court without filing with their insurance company. But that is a tremendous hassle and situations like these is one major reason to carry insurance - the insurer can figure out whether it is worth pursuing compensation if the offender is uncooperative.
OP - thanks for posting this. I really haven't played around with sentry mode and didn't think I'd bother with it until I saw your post. Will definitely be using it when parked in public lots now.