[Someone]

The intuitively more logical “average absolute error” doesn’t necessarily have a unique solution. For example: if your samples are x1 and x2, any estimator between x1 and x2 has minimum average absolute error.

Squared error doesn’t have that problem (the midpoint beteren x1 and x2 uniquely minimizes it) and (very important historically) is easy to compute for the linear regression case. That’s why linear regression and squared error won. The rest, I think, is gravy. If absolute error were easy to minimize, we might even have found/invented some other properties that ‘show’ why that is nice.