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.
Page 182 of this excerpt from Acemoglu's Economic Growth discusses aggregation and the possibility of a representative household. http://users.econ.umn.edu/~guvenen/DaronBook1.pdf
Aggregation of firms (and production functions) works quite nicely theoretically. Unfortunately, the same is not true for household / demand aggregation. The Debreu-Mantel-Sonnenschein Theorem aka. the "anything-goes" theorem gives a stark negative result. Utility functions are just too idiosyncratic to aggregate in every possible case.
However macroeconomists still do this for purposes of tractability. We know that sometimes aggregation is possible, if utility functions take certain forms. This is only one of many dimensions modern macroeconomic models are far from realistic. This is no secret to anyone.