Is $\sigma_{1} = \frac{\sigma_{\tau}}{\sqrt{\tau}}$ suitable for volatility scaling?

Question

It seems to be the de-facto method; and I see how we get it from log-normal assumption.

However volatility scaling seems to be way more sensitive to $\tau$ than mean scaling -- as in two ~ 2 times (be it using simple or log returns) in my sample.

And it indeed it is: \begin{equation} \frac{d}{d\tau}\big(\sigma_{1}(\tau)\big) > \frac{d}{d\tau}\big(\mu_{1}(\tau) \big)\hspace{2.5 mm} \mid \hspace{2.5 mm} \tau > 0 \end{equation}

But is this good for practical use?