That's not even a minor problem. And glancing through several different presentations of free markets, it's not even a common assumption. The closely related mobility of factors is a fairly common factor (that is, if you don't like X, you can readily find Y to replace it) which can be broken by natural monopolies.
That is the initial neoclassic response to the problem of unemployment. The condition that there is no production fixed cost was added to the neoclassic theory later (Kenneth Arrow & Gérard Debreu). You can easily understand the problem: in neoclassic theory, demand and supply meet each other at the optimal point. But if there are fixed production costs, producers cannot stand adjustments that drop the market price below that cost.
As I said labor comes with a fixed cost, as it needs workers to stay alive and be renewed. Latest neoclassic theory therefore teach us that labor cannot be a free market. As a consequence a market that involves labor will be "tainted" by that problem too.
[if the labor market makes workers too poor, you get an overproduction crisis] But not in the real world.
Do you mean overproduction crisis do not happen, or that they are not caused by a weak demand?
Again, you're telling me how bad free markets are for a market that isn't even close to free. It'd really help, if you had experience with a free market.
I do not deny that some free market exist, and that neoclassic theory predicting maximum efficiency may have some ground there. Wholesale raw material market is an example. I just disagree with the idea that this theory can be applied to any market. Latest neoclassic economists themselves demonstrated it was not true.