If you remove the independent variable term, βx, leaving just the model y = α+ε, and then solve the expressions to minimize the sum of the squares of errors and sum of absolute errors (calculus is quickest, but not totally rigorous for SAE; differentiate wr.t. α and set the derivative to zero) as a function of α you will find, fairly easily, the results that the optimal values under which each objective is minimized are the mean and median respectively. The prior work shows that these objectives map to MLE under the respective distributions. The heuristic conclusion here is that minimizing SSE is "right" for the Normal and minimizing SAE is "right" for Laplace, if you accept that MLE is a "superset" procedure.

FYI: to do the SAE minimization one puts d|x|/dα = sgn α, which is not completely rigorous as x=0 should be considered and d|x|/dα does not vanish at the origin. It can be done rigorously to obtain the same result, but that's more complicated!