In the Byzantine Generals Problem as stated in the paper, there is a single commander that decides on the plan. Paricipants known the identity of the commander in advance. In this case signatures are sufficient.

The problem you are describing is a different one, where there is no single commander and a group of generals needs to collectively decide on a plan. There signatures are indeed not sufficient. The way proof-of-work solves it is by randomly electing one of the generals to become commander for this decision round (i.e. 'it mines the block'). The random election is somewhat imperfect, leading to second order complexities such as probabilistic finality.