That only works if 1) the other person had valid insurance and 2) they (the car owner or their insurance) actually respond to your insurance.Leave it to your insurance to "go after" the car owner. That is what you pay them for.
Last year my wife and I were in Dallas and ended up getting side swiped on I-35. We pulled over with the person who hit us, got their information, took pictures and called the cops. It was a rainy day in Dallas, so of course it was going to take forever for any cop to show up. The other party refused to wait and we had no reasonable timeline as to when the cops were going to show up. We ended up calling the non-emergency number back and telling them that we couldn't get the other person to stay and asked what we should do. They told us that we wouldn't be able to file a report without the other party. I now believe that we should have insisted that we get at least a one sided report on file. That was our mistake.
Tried to file on the other party's insurance (surprisingly was valid insurance), but the guy would absolutely never respond to his insurance. As such, his insurance said that they can not accept responsibility for the accident because they only have our side of the story. Filed with our insurance and had them go after his insurance. Got our car fixed and was out about $1000 in deductibles and car rental that should have been reimbursed when they got money back from the other person's insurance. Lo and behold, he wouldn't respond to our insurance either. Eventually our insurance told us that we were on the hook for the deductible and rental because they couldn't get prove that it was the other person's fault. I bought a dash cam that day and installed it that weekend. I will never leave an accident again without a police report and I will do absolutely anything I can to make sure the other party keeps their slimy 8ss there until the report is filed.
Even now, over a year later, just writing that out makes me want to track down every single person that was involved in this that screwed us over and pour alcohol down their urethra.