Why can't charge be in a stable equilibrium in electrostatic field?

I am reading Feynman's Lectures volume_II where he discussed the impossibility of the presence of stable equilibrium in an electrostatic field.

There are no points of stable equilibrium in any electrostatic field—except right on top of another charge. Using Gauss’ law, it is easy to see why. First, for a charge to be in equilibrium at any particular point $P_0$, the field must be zero. Second, if the equilibrium is to be a stable one, we require that if we move the charge away from $P_0$ in any direction, there should be a restoring force directed opposite to the displacement. The electric field at all nearby points must be pointing inward—toward the point $P_0$. But that is in violation of Gauss’ law if there is no charge at $P_0$, as we can easily see.

Fig. 5–1.If $P_0$ were a position of stable equilibrium for a positive charge, the electric field everywhere in the neighborhood would point toward $P_0$. Consider a tiny imaginary surface that encloses $P_0$, as in Fig. 5–1. If the electric field everywhere in the vicinity is pointed toward $P_0$, the surface integral of the normal component is certainly not zero. For the case shown in the figure, the flux through the surface must be a negative number. But Gauss’ law says that the flux of electric field through any surface is proportional to the total charge inside. If there is no charge at $P_0$, the field we have imagined violates Gauss’ law. It is impossible to balance a positive charge in empty space—at a point where there is not some negative charge. A positive charge can be in equilibrium if it is in the middle of a distributed negative charge. Of course, the negative charge distribution would have to be held in place by other than electrical forces

I am having problems with the explanation especially the bold lines. It can be comprehensible that to be at equilibrium at $P_0$, the force i.e. the field must be zero. There must be a restoring force in order to bring back the charge at $P_0$. Feynman then wrote that as there is no charge, it cannot have any net flux as is evident from the bold lines. Confusion! Where did the charge go that he was discussing so far? He only displaced the charge , previous at $P_0$ away from $P_0$. That doesn't mean the charge is no more inside the surface!! I am really not getting what/why he wrote if there is no charge at $P_0$. Can anyone help me clear this confusion?

Also, if the field is zero, doesn't it mean that flux is also zero? If not, why?