If Hertz has imvoiced the cost of damages, would that not meet the needs of an estimate since it is the actual charge and not just an estimate

This is normal in Europe ime

surelty, Hertz doesn't have a demand letter since it is already paid and nothing to demand

Well, those are the requirements. In a recent claim (for a similar windshield crack) Alamo had no issue providing the required documentation. The claim was settled quickly. Hertz is ignoring everyone. Hertz certainly knows the requirements here in the US and could easily provide whatever is needed.

" Hertz certainly knows the requirements here in the US and could easily provide whatever is needed."
I think that's an incorrect assumption. The 'operational' side of the business is local to each country. I doubt that the fleet management staff at Hertz Switzerland know or care anything about how US insurance companies operate. Your contract was with Hertz in Switzerland, and the small print on your rental agreement would show that the contract is governed by their laws.

It seems an UScentric view that Hertz in Switzerland should know ( or even care ) what one particular insurance company in the US is going to want - the world does not revolve around the USA

Do you expect every car hire company in every country to know what every other countr will want and to pander to them?

As mentioned, the rental was with a Swiss company under Swiss law. It has given you the breakdown of what the costs are and charged accordingly and has provided a receipt. It seems to have done what it needs to do

The issue , to me, seems to be the company through which your insurance is provided not being able to recognise that other countries operate differently to the US and to handle it accordingly

I would send an email Hertz Switzerland customer relations and see whether it can assist in providing what you would like - I would also be looking at another company for providing rental car coverage if your current one will not accept an invoice as evidence of the cost

I think you have me confused with the OP.

And the CDI / TDI logo, glowplug dash light and the putt, putt, putt of the oil burner engine didn't give it away? Even Audi's & Merc's best diesels still sound like diesels.

I don't know that I'd call that Hertz negligence. American negligence maybe, but not Hertz. I agree that it can be confusing, especially with the green nozzle NOT being diesel in non-US countries, but that still isn't Hertz' fault. Hertz rents cars to people from many countries with many different conventions in driving and I can't expect them to know the laws and customs of my country vs. theirs.

I don't know what CDI/TDI means, I don't know where glow plugs are and are not used (but I know what they are), and I don't know what a diesel engine sounds like. Not everyone has a mechanical background.

If I ever rent a diesel, they better hope there's a very obvious sticker somewhere

Update:
Hertz did not respond to Amex's request for information so Amex removed the charge. I just received a bill directly from Hertz in Switzerland. I guess I will reply and ask for a copy of the repair bill.

I am worried they will screw up my otherwise perfect credit.