This is the right answer but maybe not to the right question. For most applications the linearity of variance does not really matter (e.g. when taking descriptives of a sample), at least in the way they are used/taught. Not that many appliers of statistics do computations where the mathematical properties of variance have any significance, and even when it does (e.g. ANOVA), most appliers don't really understand where it comes from.

I think the OP's frustration in how computation of standard deviation as a quantity is just taught by rote in a cookbook manner is quite understandable. If one's not thought more in depth about distributions and estimation of their parameters, it really makes little sense.

Whether the statistics should be taught more in depth is another question. The current state of the mathematics of statistics and probability is such that it needs quite heavy tools to do rigorously. And countless of hours of getting to know the quirky inconsistent notation.

But until it is taught more rigorously, more intuitive measures of dispersion like MAD make a lot more sense for most applications. And I don't think it would be an unreasonable ask for the statisticians to lay out the "porcelain" for working with the more intuitive, if somewhat more analytically inconvenient measures. A bit like we don't require programmers to understand how transistors work.