For the QB, not a huge difference between a properly inflated and a deflated ball, but for the receiver of throw, it does make a difference. A harder ball can deflect from your hands easier and is harder to grab, than a softer ball. If you have played any sports where catching objects with your hands is involved, you would know this already. I played cricket during my youth, and a well worn ball was certainly easier to catch.
I actually have caught (and thrown) quite a few footballs in my time. I don’t know that the less inflated ones make a difference.
Without facts to back up either position, we’ll just have to disagree.