I don't know about Hertz and Avis (as we never use them in Florida) but Dollar, Alamo and Budget just are not interested in points, unless something's changed in the last few months. Whenever I've asked, I've just been told what I quoted above 'if you're qualified to drive in the UK, then you are qualified to drive in the US' from car rental companies.

My husband had points on his licence and we never even were asked about them when renting a car many times from the above companies, despite the fact that they were clearly marked on his driving licence. which was shown to the car rental desk.