What software is most often used by these persons? I assume pico8?

I love the way you describe the time travel aspect. My own project has to run on an NES, and memory constraints mean I can't actually store a previous game state entirely, so there are endless hacks and cheats to "rewrite history" after resolving most of the enemies in advance, to account for the player's actions at the last moment before display.

I can't cheat the beat timing though (drawing is way too slow) so I've got the beat judgement tuned to look less awful while being reasonably forgiving about late inputs. To compensate, I stretch out the animation timing on the following beat, so the early cels aren't weirdly bunched up after late inputs. It works well enough given the constraints.

It's fun seeing all the different approaches to this problem in the rhythm game sphere! It's way more complicated than you'd initially think, and games have all sorts of tricks up there sleeve to deal with it.