Markov Games (MG) is an important model for Multi-Agent Reinforcement Learning (MARL). It was long believed that the "curse of multi-agents" (i.e., the algorithmic performance drops exponentially with the number of agents) is unavoidable until several recent works (Daskalakis et al., 2023; Cui et al., 2023; Wang et al., 2023). While these works resolved the curse of multi-agents, when the state spaces are prohibitively large and (linear) function approximations are deployed, they either had a slower convergence rate of $O(T^{-1/4})$ or brought a polynomial dependency on the number of actions $A_{\max}$ -- which is avoidable in single-agent cases even when the loss functions can arbitrarily vary with time. This paper first refines the AVLPR framework by Wang et al. (2023), with an insight of designing *data-dependent* (i.e., stochastic) pessimistic estimation of the sub-optimality gap, allowing a broader choice of plug-in algorithms. When specialized to MGs with independent linear function approximations, we propose novel *action-dependent bonuses* to cover occasionally extreme estimation errors. With the help of state-of-the-art techniques from the single-agent RL literature, we give the first algorithm that tackles the curse of multi-agents, attains the optimal $O(T^{-1/2})$ convergence rate, and avoids $\text{poly}(A_{\max})$ dependency simultaneously.