Yes, the royal family is becoming more receptive to public opinion because they see how public opinion is affecting the lives of those in charge. Clearly there are problems within the country and the only way to fix them is by starting a dialogue between those in charge and the people.
They clearly executed him because public opinion swayed against him and against them. Whether they are swaying to public opinion because they are afraid of what the public will do, or because they have started to have more compassion towards the public, it is clear that something has changed in the royal family.
Royal families are only still around as an act of tradition. They usually do not have any real political power. People just like the idea of royalty. The public is allowed to be critical now if they so wish because they need not fear punishment for speaking out or holding the royal family to normal people standards.
Saudi Arabia likes to pretend that it is a modern and free country. That is not the case. Most of the 9-11 hijackers were from Saudi Arabia. They still have a great deal of backwards thinking in that country. Part of that is allowing the royal family to do whatever they want to. There is no accountability. That execution was just for show.