This was super fun! We used the same concept for our game, restting in time and solving puzzles with past versions of yourself. But yours is a slightly different twist where it's like a speedrunning challenge with a set number. It's really really fun, and I really like how you can use more runs to optimize. Sometimes the old ghosts get a little off pattern---which makes sense if they hit a door or something---but sometimes it's without reason, which can be a little frustrating when you are relying on them to follow a pattern.
But your 'past' selves also function different than how we did ours, which explains the ability for them to do new things. I think that's something that can be explored more too, having puzzles where you make a past self do something different by altering the world to solve a puzzle?
I'm rambling, but super fun game, and really well polished.