It will not cause more lag, it will just not be accurate to where your cursor actually should be when the frame gets displayed. For people with strong computers you would only need to predict a few ms forward in time which should be possible to get really accurate.
>it can never predict when you will hit the button to fire your weapon
That is not what is happening. What is happening is that time between registering the click and the frame being displayed. The prediction I'm talking about would be predicting subframe mouse movements. This behaviour is more for smooth mouse look. For shooting you want to have a precise timestamp of when the click happened so that you can see if at that moment in time the shot would have hit. The position where you are looking when shooting and the position where you are looking on screen are different things.
That's impossible, if you change your movement it will mispredict and thus cause more lag, while exactly when you change is when it should feel the best. Consistency feels better.
Just like no magic gizmo can reduce audio lag other than have actual proper low latency standards (for bluetooth audio): it can never predict when you will hit the button to fire your weapon, etc..., in games