This is a big problem, which is enough to kill the model, in my view, but not the only problem, and not the one I alluded to. (And IMHO it affects supply as well.) The problem I alluded to is that how you aggregate supply and demand curves so that the equilibriums (the points where they intersect) match in some reasonable way.

Supply and demand curves are such an unsalvagable mess, in my opinion. Economists should just stop drawing them.