For example, you find a way to swap two pieces on the top layer and mangle the bottom (f), turn the top (g), and then do the opposite (f^-1), swapping a different pair and un-mangling the bottom. Between complementary swaps, edge flips, and corner rotations, you can build an entire solution with this technique. (My current version of this does the edges first, ignoring any damage to corners and then does corners.)
Somewhat related - many years ago there was a tutorial of the Gap computer algebra system that analyzed the rubik's cube group. I can't find the original, but there is a translation to Julia here: https://oscar-system.github.io/GAP.jl/stable/examples/
You eventually develop the intuition to solve any move without having to "memorize" anything.
(ps.: after a quick search I see that one can buy replacement stickers for a few bucks on Amazon :D)
Hm, so something akin to a bidirectional path-finding problem, where one can still call it "brute force" because both known positions (start and goal) are each doing a breadth-first search, as opposed to something fancier than picks a direction.
Can a Rubik's Cube be brute-forced? - https://news.ycombinator.com/item?id=36645846 - July 2023 (108 comments)
(Reposts are fine after a year or so; links to past threads are just to satisfy extra-curious readers)