I know this might be controversial but I always hated the way time travel was portrayed in Prisoner of Azkaban. It’s this endless time loop where Harry and Hermione change absolutely nothing, they just gain a little bit of perspective.
By removing Hermione throwing that stone and Harry seeing “his father” at the lake you could make it look like it’s the classic way of time travel. It would make moments like the saving of Buckbeak more meaningful aswell because they actually achieve something.
Though you would have to find solutions to Harry and Hermione noticing the minister and Dumbledore coming (maybe add a footstep sound effect) and Harry getting saved at the lake.
I’d do this for the simple reason that I don’t believe closed time loops are logical.