Bar Keepers Friend has been recommended to me a number of times, I recently got a chance to try it out after turning on the wrong burner which just happened to be under a mostly empty pot leaving scorch marks. (This happens to me a lot, but this time I have nice enough cookware that it didn’t outright melt.) And wow people, the most lazy of scrubbing with Bar Keepers Friend and a regular sponge and it nearly completely restored the pot. Thanks for the recommendation, high fives all around.
update: Marielle left a very important note — wear gloves when using this! It is not something you should get on your skin.