Probably not. In any event, rental in your home country is only covered if you have booked a hotel or flight in conjunction with the same journey - an exclusion which Amex does not point out in the 'big print' in the brochure referenced above.

So, for instance, you are not covered if you rent a car for a day to do a day trip. More interestingly, you are not covered if you rent a car for the weekend to drive to your parents house where you will sleep over. However, if you drive to your parents house but choose to stay in the hotel round the corner, then you ARE covered. If you FLY to your parents nearest airport in the UK and then rent a car to do the last few miles, you ARE covered if you stay in their home, though! Clear enough?!