The rental company is legally obliged to have third party cover on the car. If you book through a european website, that will be included in the quote. Book from a US site and it might not be (but you'll still be charged for it - no way to avoid).
You'll need to check the wording of your AMEX policy. They don't have a global policy - it depends on the terms of the card. My card does provide cover, and it doesn't matter that the basic cover is provided by the rental company. Amex just payout for any additional cost (the excess). If you do have coverage through a card, note that it does not work in the same way as in N America. The rental company will never deal with your card insurer. If you have any damage, they will charge you and it is then down to you to reclaim this from your card policy. Slight hassle, but it works. I've had a few claims paid by Amex, never had any quibble.
For simplicity, you can pay the rental company for a "zero excess" coverage. But this is the expensive option, this option can easily cost more than the basic rental.
Cultural difference. All European rental companies are strict on damage control, to a degree that shocks some customers from N America. You should examine the car thoroughl for any existing damage before you drive away and don't drive away if there is any existing damage which is not logged on your agreement.
Never rent from Green Motion - just put that into any search engine and read the endless tales of woe on damage claims with them.....