There is a simple answer to your question. To take the dalet minim or make a bracha on the sukkah would be a zilzul (disgrace) to the independant yom tov of shmini atzeres beacause you are clearly declaring the day to be sukkos in some way. On the other hand, eating in the sukkah is not considered a zilzul because people will eat outdoors from time to time. Therefore, doing so does not necesarilly indicate that we are still holding in sukkos. (pe’amim shusukaso areiva alav.)
That being said, when living in a cold climate (i.e. Poland or Russia) some did consider it a zilzul to eat in the sukkah because no one would ever eat outddors in such conditions if not for the mitzvah of sukkah. Hence the minhag of many communities was born to not eat in the sukkah on shmini atzeres.