Where am I wrong in thinking this way?
The error is that it's not stable. Stable equilibrium means positive eigenvalues of the system's Hessian. Alternately phrased, stable means that any small push of the system out of equilibrium will eventually relax, and the system will be in its original state. However, in the example you gave, a small push of the positive charge will result in the charge uncontrollably accelerating towards the metal surface.
So, it's not stable.