I don't know what documentation would be needed for the Admin fee. If it was stated in the contract, a copy of the contract should be enough. If not, then how do they document and admin fee? They are the ones charging it. How would you "document" it, other than sending a bill for it. The "loss of use" fee is a bit different. I thought I remember seeing somewhere that indicated, at least in some places, where they could not collect it unless they could show the car would have been rented during the time period. A windshield replacement doesn't take very long. In this situation, I think it's a rip off, even at only $12.50. However, given the choice of a bad credit mark or $62.50, I would pay the $62.50, and then pursue the issue if I felt it was worth it. If the CC issuer asked for proof of charges, and the car company didn't provide it, then that mihgt be the last time I used that car agency, depending on how customer service later handled my complaint.
From the OP's story, I do NOT feel this is AMEX's fault, but Thrify's.
It should be noted, that no matter what credit card coverage you have, the responsibility for payment of damages is still your's, as far as the rental companies are concerned. However, they do have to assist you in getting reimbursed, and Thrify did not.
Consider this as part of the cost of getting the lower price or other benefit you got by choosing Thrifty. My last Thrifty rental was at SNA. The online price was quite a bit lower. I went to the rental lot, where 2 employees were doing absolutely nothing business related. Because I am not a member of the frequent renter club, I was sent back to the baggage claim area desk, where there was a nice long slow line. The other companies had either few or no customers in line.
National gave me a rate as a walkup that was only $2 more per day (2 day) than the Thrify rate. I had no wait in line at National. It will be a long time before I consider Thrifty again.