Balanced Network max flow review
The flow defined by you, satisfies both the constraints: a) conservation of flow at each vertex -- by the definition of balanced network and b) the capacity constraint because $f_{e} = c_{e}$ for every $e \in E$. Therefore, it is a valid flow.

It is a max-flow because it is sending the flow of value which is equal to the capacity of the cut $C$ defined as: $\{s\}$ and $V \setminus \{s\}$, i.e. max-flow = $\sum_{(s,u) \in E} c(u,v)$ since there is no incoming edge to $s$. Note that any flow can not exceed the capacity of this cut. This fact follows from the min-cut max-flow theorem. i.e., max-flow $=$ min capacity of an $s$-$t$ cut $\leq$ capacity of cut $C$. Therefore, what we have is a max-flow.