It's unfortunate that more theory makes determinants really intuitive, but most people don't get there. The upshot is that there's a thing called the exterior algebra which is relatively simple to define and calculate with, and essentially encodes the notions of areas, volumes, 4-volumes, etc. over a space. Every linear map on a space uniquely specifies a map on the exterior algebra (in fact this is a functor, which makes calculations easier), and the determinant of a linear map is then just the corresponding exterior algebra map evaluated on the unit n-volume. The minors are the exterior algebra map evaluated on the unit volume for various subspaces.
The rules for the exterior algebra and the fact that this is a functor let you learn a couple simple rules to simplify expressions into a standard form, and then you just apply those rules mechanically and don't have to remember minus signs or what multiplies with what. It becomes a process that requires no thought.
It's sort of like how once you learn how to deal with complex exponentials, you can forget pretty much every rule from trigonometry, making it entirely pointless to memorize those rules.
The rules for the exterior algebra and the fact that this is a functor let you learn a couple simple rules to simplify expressions into a standard form, and then you just apply those rules mechanically and don't have to remember minus signs or what multiplies with what. It becomes a process that requires no thought.
It's sort of like how once you learn how to deal with complex exponentials, you can forget pretty much every rule from trigonometry, making it entirely pointless to memorize those rules.