Plus, does "not... but" mean "not only... but also" here? It could, yes.
But never forget that the true character of our nation is revealed not during times of comfort and ease, but by the right we do when the moment is hard. Here, "right" means the "correct things" we do when times are hard. "When the moment is hard" is not something a native English speaker would say. The adjective "hard" is not appropriate for the noun "moment" in this context.