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We study the problem of planning restless multi-armed bandits (RMABs) with multiple actions. This is a popular model for multi-agent systems with applications like multi-channel communication, monitoring and machine maintenance tasks, and healthcare. Whittle index policies, which are based on Lagrangian relaxations, are widely used in these settings due to their simplicity and near-optimality under certain conditions. In this work, we first show that Whittle index policies can fail in simple and practically relevant RMAB settings, even when the RMABs are indexable. We discuss why the optimality guarantees fail and why asymptotic optimality may not translate well to practically relevant planning horizons. We then propose an alternate planning algorithm based on the mean-field method, which can provably and efficiently obtain near-optimal policies with a large number of arms, without the stringent structural assumptions required by the Whittle index policies. This borrows ideas from existing research with some improvements: our approach is hyper-parameter free, and we provide an improved non-asymptotic analysis which has: (a) no requirement for exogenous hyper-parameters and tighter polynomial dependence on known problem parameters; (b) high probability bounds which show that the reward of the policy is reliable; and (c) matching sub-optimality lower bounds for this algorithm with respect to the number of arms, thus demonstrating the tightness of our bounds. Our extensive experimental analysis shows that the mean-field approach matches or outperforms other baselines.