I don't argue with Left's principles of fly casting which are independent of casting style. But when he says it is best accomplished by one style of casting, then it becomes debatable.
How To Fly Cast- Fly Fisherman
For example, several of the best know instructors and the arguably the best fly cast in the USA use a style different from Lefty.
"This is the arm style of people who have most influenced casting in California, including Jimmy Green, Mel Krieger and Steve and Tim Rajeff. "
Al Kyte discusses the various grips in the article above. He says that it may look like casters start with the thumb on top grip, but as they cast; they switch to the key grip. This is exactly what Steve Rajeff does.
The thing I noticed on Steve Rajeff's video below is that he does NOT use the thumb on top grip that most newbies are told to use. Look at the slow motion at 50 seconds and 1:26 on the video below and he is using the "key" grip (he holds the rod grip as you would hold a key). It is called the palm out grip in the the article by Al Kyte above.