Overall an exciting idea, can't wait to see how it turns out! I was immediately worried about the technical side of this as there can be a few gotchas in trying to get the echo mechanic to work, but really happy to see you've prototyped up the idea. There still might be some problems particularly if you rely on physics (do you then record the movements of all effected objects? physics isn't quite deterministic in Unity), and there may be some paradoxes that you need to sort out where what you do with the current player is in conflict with the consequences of the echo (even a simple example is what happens if you are standing inside the door when the echo closes it by getting off the switch? this might be easy, but what may be hard is predicting all the possible paradoxes).
Other than that, well written and justified! Needed more details on what the puzzle/environment aspects actually will be and what the player looks like but great job.