I've had the experience twice. Both times in Europe (I'm an American) so CC coverage was primary.

The first time, I used a VISA card and rented from Europcar in Austria or Germany. I filed the report and VISA took care of the entire thing. I didn't pay a penny and didn't have to follow anything up.

The second time, I used an Amex card and rented from Avis in Switzerland. On this one, the start of the process was similar, except there were a number of charges that Amex said they didn't cover, such as loss-of-use, and maybe a couple of other things. At Avis' rates, these items came to about $500. Avis sent me a couple of notices saying I owed the money, but I never got around to paying them and I haven't heard from them for some years now.

I don't know if those comparative examples really say whether VISA or Amex is better, though. The difference could be entirely between Europcar and Avis, or simply between the two situations.

I have not tried to rent a car from Avis in Switzerland since the event listed above, but I wonder if I could expect an unpleasant reception if I did. I have been back to Switzerland recently, though, so I assume they at least did not put me on some kind of eternal Interpol list or something.