Hyde White was more prominent in Hollywood than Coote (who was quite prominent on Broadway though) and we all know how much Jack Warner completely disregared theater talent. Imagine, Cary Grant as Higgins and James Cagney as Doolittle. My God. What an awful film it might have been!
At least Cary Grant had the good sense, and wit, to turn it down: "Not only will I not be in your film, but if you don't cast Rex Harrison, I won't even go see it."