Why power and not amplitude?
It's generally not true that the average deviation from the mean is not important or much less important than the power of the deviation. For sampled data, the mean deviation would be defined as

$$\epsilon=\frac{1}{N}\sum_{n=0}^{N-1}|x_n-\mu_x|\tag{1}$$

where $\mu_x$ is the mean of the data. This quantity is definitely relevant. However, the absolute value in $(1)$ makes it much less convenient to use than the equally relevant standard deviation

$$\sigma=\sqrt{\frac{1}{N}\sum_{n=0}^{N-1}|x_n-\mu_x|^2}\tag{2}$$

which is differentiable, and, consequently, easier to handle analytically (such as in least squares fitting, etc.). (Note that in Eq. $(2)$ you might as well use a factor $1/(N-1)$ instead of $1/N$, but that is not relevant here).

Apart from the above argument based on mathematical convenience, it is true that in the case of uncorrelated noise sources, it is the noise variances (i.e., the squared standard deviations) that add up, and not the mean deviations.