No, the initial reps with lower weight are most useful for ensuring, that you perform the exercise with sufficient quality.
E.g. for squats focus on maintaining a very high tension in your lower back and the deep abs. Look in the mirror and check that your knees do not moves during the exercise.(vertical shins)
They work as a safety precaution too: You get an opportunity to feel if your body is not ready for max weight in the exercise.
Perhaps you did not notice a small injury, which could be fatal if you perform the exercise with max weight. Often injuries first show up at e.g. 50%.
It is common to go even much lower than 60%, and for squats your initial step could be (jumping) air squats (0% weight) as you suggest.
All trainers I ever had, always has told me to start lower and use smaller steps than I think I need myself. It does not means more reps: When you get closer to max, you just need one rep.
When I perform squats, my first step is cardio for heating up the muscles and bringing fluids to the joints. Then static hindi squats for ensuring mobility and range. That means you can go deep in your squats, but you will also get a few seconds to be aware of "changes" in your body.