I just got done reading Gorard's "Revisiting a 90-year-old debate: the advantages of the mean deviation." (https://www.leeds.ac.uk/educol/documents/00003759.htm) I'm no expert at statistics; in fact, I'm quite aware of how much there is that I don't know. So maybe this question is actually silly, but I'm gonna give it a go because information regarding this seems to be sparse.
Gorard talks a lot about how the MAD doesn't lend itself as well as the SD to algebraic manipulation. My question is this: are there pieces of statistics which could not be essentially rephrased using the MAD as the concept of spread rather than the SD? Has anybody even tried? If so I'd love to check out their work! I expected this to be an endless rabbit hole of alternative statistical theory but I haven't been able to find much on it.