I can't speak specifically about Hertz, but my wife had an accident with an Avis car, and we pretty much followed the same steps, except we filled out the Avis form in person at a local location where we got a replacement car.
I filed claims with both my insurance and Amex. It took Avis almost 5 months to get the car evaluated and submit the paperwork to my insurance. The insurance paid less the $500 deductible, and Amex paid that.
So it took a while for everything to happen, but no out of pocket expense and no real problems getting it resolved.