Final answer:
Functionalism is the theory that implies society would face disruption if the elderly vacated their roles only upon death or incompetence, emphasizing the importance of their orderly transition for societal stability.
Step-by-step explanation:
The theory that suggests society would be disrupted if the elderly left their positions only when they died or became incompetent is functionalism. Functionalism is a sociological theory that focuses on the functions or purposes of various elements within a society and how they contribute to the stability and continuity of society as a whole. According to functionalism, each part of society is interdependent and contributes to society's functioning and stability. The roles that the elderly play, their exit from those roles, and the timing of that exit are all considered to be significant to the smooth operation and stability of society.