In order to minimize the excess demand functions we must assume that all excess demands are functions but if this assumption fails, then we can no longer use this method to find our p* equilibrium price via this method. So if the minimization can’t be done then a walrasian equilibrium can’t be found.
If we can use the method of minimization of excess demand what we are trying to accomplish is to find a price at which markets get really close to clearing. If a walrasian equilibrium exists then our p* from our minimization problem will coincide with the walrasian equilibrium. But there could be cases where the walrasian equilibrium doesn’t exist in the first place, this could be for many reasons such a a violation of u monotonicity or u continuity. The book mentions that when an equilibrium doesn’t exist we could reach an approximation via shortaging the resources or wasting some of them in order for the markets to clear.
Therefore if the general assumptions of a walrasian equilibrium are fulfilled and we are allowed to make use of rationing and wasting of resources we can always arrive at a walrasian equilibrium aproximation