Spoiler alert, it works. Details below.
I started using Amex CDW at least a few years ago. It was available on the old Costco Amex cards, now available on several versions of Amex cards (for me now Amex everyday cash card). There are some variations depending on your state of residence (some states don't let Amex offer it because it's considered an insurance product). Amex charges a one-time charge when you rent your car, roughly $25 for me. It covers up to a 40 day rental for a charge that's not much different than paying only one day with the rental company - great deal.
Over the last handful of rentals, both in Europe and domestically, the only issue I had was with Autoeurope. Amex didn't pick the charge up automatically. I had to call them and have them manually add the premium charge to ensure I was covered.
In June, I had my first accident. It was pretty minor but, as you can expect, still resulted in about $3,000 damage. I just got word yesterday that Amex paid the claim. They didn't pay an administrative fee of $138. I expect another bill from the rental company for that, which is relatively minor - maybe they'll accept the payment they received and call it good?
I did have a couple issues with Amex. First, they want you to call as soon as possible. It would have been much more convenient to let me report the crash by either email or even online chat, but they don't give me that option. When I did call, they told me I could file the claim online, something that would have been more helpful if I had known it earlier.
The online claim worked reasonably well. As expected, they wanted details of the accident. Again it would have been good to know what they wanted earlier. I normally photograph the car pretty thoroughly to prove I did no damage. In this case, I dropped it off and didn't take photos, figuring I didn't need to prove I didn't damage it. Like a bonehead, I didn't think that the insurance would want to see proof of damage. Fortunately, they accepted an internet image of the car model. I detailed, on the photo, the damage to the car I'd noted. I had all the other information they needed.
The only other issue: I sent all the information, in attachments, in a reply to the person who said they were on the claim. Two weeks later I got another email saying they still needed the information. I went back to the online claim and put in electronic copies of what they needed. I was a little surprised my reply with attachments didn't work. After I put all the attachments in, the next thing was notice that they had paid the claim.
Last tidbit. This was in France. I hit a big truck that suddenly stopped halfway into a roundabout. I don't think he realized I hit him, it wasn't much of a crash. I followed expecting him to pull over but he never did. No one ever asked me for my IDP, the rental company, the insurance company, police weren't involved (but I had it if they did).