Not sure if that's the best solution to the strengthening mechanic, but I can't think of anything either.
Having the option to auto move is definitude a good quality of life update. I just tested it and feels pretty good. I think it would be cool if you could speed up slightly by fast tapping, like hypertapping in Tetris (big Tetris fan, so I can't help using it as an analogy).
Keep up the good work!