So they can offer all they want, but no one is asking. Firms have their workers, and except for replacing retirees, etc., they aren't looking for new workers. So the fact that the involuntarily unemployed are willing to work for less does not mean that the wage rate will fall. That's the critical difference in Keynes' view. When you think about the fact that people aren't re-hired every day, suddenly the vision of the masses of potential workers competing with one another, driving wages down, doesn't make sense.
Firms have their employees and they are content.
The existing workers would be willing to offer to work for less if it meant gaining employment, but they already have jobs. The unemployed would gladly offer to work for less so that they could find work, but no one is hiring. Involuntary unemployment is not an unstable situation. We could remain there forever. There are other reasons why the Classical version does not make sense.