I want to write a counterargument, but first I need to know what type of game you are talking about.
If it's a FPS, then the mouse has the benefit of sensitivity, since it takes less to orient yourself than a controller. However, the analog sticks have the benefit of knowing how fast you are moving, since you can only move the sticks so far. I'll acknowledge that this can be resolved by tweaking your sensitivity options.
Secondly, I don't understand this:"Wait for your character to rotate in that direction", for two reasons. First, this happens with both analog sticks and keyboards. Second, that is why you don't just have a mouse or one analog stick: the keyboard/other analog stick is used to move, which can also be used to change your camera angle faster/compensate for the direction change by moving. Anyone who has played an FPS knows that you can't stay still; that leaves you open to attack (unless you are a sniper...)
Ultimately, the choice between analog sticks and mouse + keyboard comes down to playing style. A mouse provides speed, but at higher sensitivities, you lose accuracy. An analog stick also suffers from this problem, but it has the advantage of knowing how much movement of the stick would affect the movement of the camera (since that speed is constant).