With the rapid growth of the electric vehicle industry, the recycling of power batteries has attracted significant attention. In light of current circumstances, the question of how the government can incentivize relevant stakeholders to actively engage in recycling and improve its efficiency has become increasingly pressing. In this context, this study analyses and develops four closed-loop supply chain recycling models to investigate how different government subsidy recipients under varying power structures influence recycling efficiency, profitability, and the overall supply chain structures. The following conclusions are derived from numerical simulations: (1) Government subsidies serve to elevate recycling prices, expand profit margins, and consequently boost the volume of recycled batteries, thus incentivizing corporate engagement in recycling initiatives. (2) When the processor assumes the role of the leader in the Stackelberg game framework, it can maximize the overall efficiency and profitability of the supply chain. (3) The sensitivity coefficient and the competition coefficient are closely interrelated, exerting opposing impacts on the recycling decision made by enterprises. (4) The supply chain leader plays a crucial role in ensuring orderly supply chain development, with government subsidies of the supply chain being transmitted to its members through the leader. Consequently, this study offers a theoretical foundation for the government to enhance policy-making and for enterprises to make informed decisions. It also holds significant practical relevance in addressing the challenges associated with power battery recycling.