What is the reason why we don’t take arba minim on Shmini Atzeres in Chutz La’aretz if the SA paskens to yes eat (and possibly even sleep in the Sukkah according to some poskim) albeit without a bracha?
The Beis Yosef (651; 668) mentions that taking the minim on Shemini Atzeres is a zilzul to the Yom Tov, also stating that it is a ‘weekday matter,’ and also mentioning the issue of muktzeh. The idea that emerges from this is that Chazal did not wish to enact something that will be unfitting for Yom Tov.
It is also possible that the order of Sukkos is that on hoshana rabba we put down the lulav and esrog, and take the aravos (and beat them). If we would take the arba minim again on Shemini Atzeres, this would contradict everything we did on Hoshana Rabba, and ruin the internal order of mitzvos for the festival. Therefore, there is no enactment of taking the arba minim on Shemini Atzeres.