What impact does it have on Brinker's desire to expose the truth?

Are you referring to war? I think the war was an external event that the boys used to help define themselves as men. They interpreted the war as a way to prove their masculinity and self-worth. They all talked about joining, especially Brinker who felt like he needed to prove something. Brinker wanted to call out Finny's injury, which would prevent Finny from fighting. This was an attempt to emasculate Finny in front of the other boys and blame Gene.