Apart from Weber's model, it is not "the transfer of authority" in and of itself that promotes political and social stability, it is the manner in which that transfer of authority is conducted. A violent revolution that results in the installation of an autocratic or dictatorial form of government may provide a measure of stability, but, to paraphrase Mao Tse-Tung, only at the end of the barrel of a gun. A transition of authority that occurs under constitutional provisions allowing for a peaceful succession, such as exists in the United States, provides for social stability and freedom.
What set the American Revolution apart from revolutions and rebellions that have occurred elsewhere throughout history was the vision of a democratic society held by those who drafted the U.S. Constitution, including with regard to the institutionalization of a peaceful and orderly transfer of power -- one that enjoys complete legitimacy in the eyes of the public. Many countries have experienced revolutions and transfers of power that resulted in instability or in stability enforced through political repression. The French Revolution, which occurred in roughly the same time span as that of the American, took many years to play out before stability within a democratic system was solidified. Transfers of authority in dictatorships are sometimes marked by considerable political and social instability as possible successors vie -- sometimes violently -- for power.
Within Weber's model, the transfer of authority is important for the social stability of the state because those with authority (like everyone else) are mortal. Regardless of whether a leader has charismatic authority, traditional authority, or rational-legal authority, authority will need to be passed from that leader at some point. If authority cannot be transferred peacefully, social stability cannot be achieved.
In order for states to have social stability, the people need to know that the government will remain in power and that power will pass smoothly from one leader to the next. If they do not know that this will happen, they will live in a state of fear whenever it appears that power is going to be passed from one leader to another. Without set procedures on how to transfer power, there might be coups and attempts at coups to try to take power from a leader. Without set procedures for transferring power, a leader’s death might result in a tremendous power struggle that might turn violent. These sorts of things are very bad for social stability.
Thus, it is necessary to have set procedures for how and when authority will pass from one leader to another. Such procedures allow the government to be relatively predictable. This allows for more social stability.