Asymptotic Time Analysis for BFS Tree
You can say whatever you want as long as it is accurate.  Both $O(n)$ and $O(\text{Depth})$ are accurate, so you can say either one.  $O(\text{Depth})$ is more descriptive, since $\text{Depth} \le n$ always holds.

I don't know why you don't think it provides any insight; I think it does provide some potentially useful information.