I also find the idea of Hollywood execs deciding that they absolutely cannot change anything in the Broadway musical that they're adapting to be a bit rich coming from the same Hollywood where they never met a hit they couldn't make changes to for their adaptation.
Yes, Hollywood had been cheerfully wrecking Broadway musicals for decades--and critics and the public had been complaining about it for decades. My Fair Lady was not just any old Broadway musical. In 1964, it was considered the Broadway musical, the one to put in a time capsule, the one to take to a desert island, the one to rescue from a burning library--the one. I can't over-stress this point to people who may not have been there at the time. The moviemakers who brought My Fair Lady to the screen were determined to prove that, for once, they could do it right.