Here's the exact list of what AMEX Assurance requires from the car rental company to process a claim. Alamo in a recent claim was no problem whatsover. Hertz is AWOL:

Accident / Incident report
Photos of damage
Demand letter
Appraisal or estimate of damages
Fleet Utilization Log or proof of payment for completed repairs

In actuality, I supplied the incident report and the photos. All Hertz needs to provide is the demand letter and an estimate of the damages. The fleet utilization log is only needed if they try to claim loss of use, which I don't believe they are doing. My receipt is not an appraisal or an estimate of the damages, at this point it's just a random dollar figure that Hertz ran to my credit card.