using just the ABS gives you the LAD. least absolute deviation is not used for a couple of reasons. LAD doesn’t have a unique solution - you can code it up and your program will give you a set of solutions which will all minimize the LAD. this looks like a feature, but for various pedagogical reasons, statistians insist on BLUE. LAD can give you the LUE ie linear unbiased estimator, but not the B ie best. also, LAD doesn’t have a closed form. so you can’t write an equation and say this is the LAD, then differentiate it and derive interesting properties about the estimator. since there’s no closed form, you can use numerical methods to find something that’s good enough like error below 1e-6 and declare that to be the LAD. but that will happen at a bunch of points so no uniqueness.