After reading a great deal about the car moving incidents, and being a very happy customer of said establishment, I inquired with the staff one day about what exactly "the deal" was.
In a very matter-of-fact way I was told that, yes, they've moved cars with forklifts and that forklifts are likely to cause damage when used to move automobiles. If you park your Aston Martin and block the active/working loading areas, they have no problem picking it up and moving it so that the facility continues to function for everyone.
That simple and no-nonsense explanation made perfect sense to me... Maybe that's why I enjoy doing business with them.