From the business sense to make Alexandr Palace is not the better place. Because very small amount of tourists physically could go their. The average period of stay for the ordinary tourist group in Petersburg is about 3 days.
When tourists come to Poushkin , they are waiting in the queue for the Ekaterina Palace more for the 1 hour and then they are running through the rooms, after this, they are running back to Petersburg by bus for the next target call meal ,then to the next target in the town.
Alexandr Palace is interesting for the Europeans, but ordinary Japanese , Chinese etc they has no much interest in it because the rooms are tiny , with the lack of gold and so on.
For security reason it it very hard to protect the Eggs in the palace too.