How do balancers work in the context of counting and balancing networks?
The book is correct, and the algorithm works as advertised. See if you can prove that it does. No explicit counting is necessary.

A very similar situation occurs for languages: $(ab)^*(a + \epsilon)$ is regular even though  $\{ w : 0 \leq \#_a(w) - \#_b(w) \leq 1 \}$ is not.