1st there is not a single nash equlibrium (not to mention that it may not even be a single point but a closed trajectory in the solution space) It is highly sensitive to the initial parameters, one could at least set the initial parameters so (Rig the game) so that players eventually gravitate to the optimum eq.
There can be several Nash equilibria. But there is at least one. (Nash' theorem).
Initial conditions (and changing conditions !) will determine WHICH equilibrium is reached (hypothetically, in a steady-state situation). But what is for sure, is that no situation that is NOT a Nash equilibrium, will remain stable.
And you seem to forget that central planning ITSELF is a game, too. Who is going to be a central planner, and what are you going to do as a potential or actual central planner, IS JUST AS WELL A GAME. Central planners also optimize their personal gain.
the existence of a nash equilibrium is not a guarrantie that it is reachable. IE the earth never reaches the sun.
No central planning is not a game, it is an algorithm that determines the parameters in the economy, its gain is to find the optimum solution.