Estimating the standard deviation of a distribution requires to choose a distance.  
Any of the following distance can be used:

$$d_n((X)_{i=1,\ldots,I},\mu)=\left(\sum | X-\mu|^n\right)^{1/n}$$

We usually use the natural euclidean distance ($n=2$), which is the one everybody uses in daily life.
The distance that you propose is the one with $n=1$.   
Both are good candidates but they are different.

One could decide to use $n=3$ as well.

I am not sure that you will like my answer, my point contrary to others is not to demonstrate that $n=2$ is better. I think that if you want to estimate the standard deviation of a distribution, you can absolutely use a different distance.