It is definately not necessary for a villager to clean the beach in order to become a master builder. This is just another task which can aid in the development of a master builder. I think the confusion arises because people lose thier masters late in the game and then have to wait for an island event to train a new one (cleaning the beach).
Personally, I do puzzle #12 at the same time I'm doing #9!!
Then, I don't run into the "no more master builders" problem!!