I have had a guy at Enterprise in England crawl under the car looking for damage upon return (?) but I frequently use them in the US and never anything but a simple walk around.
I have damaged a lot of cars overseas (scraped hubcaps, seat belt caught in door caused a deep dent with paint scratched off) and had the entire side mirror busted off in Edinburgh but miraculously the whole thing snapped back into place, still fully operational and not marked up. But so far nothing made an issue even when seen by the inspector upon return. Except for once a scraped hubcap in Mexico was challenged but we had pictures to prove it was pre-existing damage. So I have experience with damage but no experience with any claims.
I used to use the Amex policy, but it has some funkiness:
You have to sign up for it in advance, and since the charge for it hits when the rental charge hits, you have no indication during rental that the policy "took" and is really in effect.
It's not good in several countries, like Italy and New Zealand. UK is OK.
It WILL NOT be activated by third party rentals like AutoEurope and maybe Priceline, so beware if paying in advance to another vendor.
If there are later charges like tolls, a one way fee, or administrative fees for parking fines or camera fines, you will get a second (or third) $25 charge on your card and when you call for what seems a simple correction you will get bounced back and forth between Amex and the insurance branch in Green Bay about whose responsibility it is to remove the extra charges. Done that circus twice.
So after these hassles with Amex and extra charges I stopped using it. There is of course no problem with the free insurance offered by most US credit cards, search this website for many successful reports of using it. Free can be better quality than paying. And always take phone pictures of the car at rental (phone pictures are easy to zoom into on the spot) in case you have an issue with pre-existing damage.
I would take a screen capture or 2 proving that the basic insurance is included in the rental and can't be waived.
Edit: I did a dummy Enterprise rental at Heathrow, and it comes up with no basic insurance. Verify that you are not choosing basic CDW in your reservation.
2nd edit: The Amex premium policy does include theft from vehicle coverage and medical insurance for occupants of vehicle after a crash.