As an important measure to involve services for older adults, the government procurement practices have become a key link for public health services. However, the information asymmetry between public health service purchasers and public health service undertakers triggers a supervision dilemma. Based on this background, this study uses the evolutionary game theory to analyze the symbiotic evolution between local governments and public health service institutions under different reward and punishment mechanisms, explore game evolution, strategy adjustment, and influencing factors of different game subjects, and analyze the necessity and appropriate intensity of dynamic rewards and punishment mechanisms. The results show that: under the static condition, the penalty can change the strategies of local governments to a certain extent, but it is still difficult to achieve complete self-discipline management of public health service institutions. If local governments implement a dynamic reward or penalty mechanism in the supervision process of public health services for older adults, the equilibrium between them tends to be evolutionary stable. For three dynamic mechanisms, a dynamic reward mechanism is more conducive to adopting a self-discipline behavior of public health service institutions, which is helpful to realize the supervision of public health services for older adults. Also, there is a positive correlation between the proportion of public health service institutions who adopt a “self-discipline behavior” strategy and the maximum punishment intensity, and a negative correlation with the reward intensity. This study provides theoretical and decision-making references for governments to explore the promotion and implementation of public health services in older adults.