However, it also has a larger purpose. Consider the fact that the only thing connecting your vehicle to the road are the tyres. Whilst that statement is obvious, what you possibly don't realise is that the total surface area of each tyre in contact with the road equates to approximately the size of a CD case. Therefore, by separating out each stage, we are only asking this small amount of rubber to do one thing at a time, rather than desperately trying to slow us down, whilst maintaining enough grip for us to change direction. The more load we place on the tyres, the more chance we have of them letting us down. If more grip is used for braking, there will be less available for steering.
Separation also involves planning. One of the common 'faults' that observers find with new Associates is that their vision is not lifted high enough. By not scanning far enough ahead, Associates can sometimes be surprised by upcoming hazards and therefore fail to properly plan for them. Hence, when approaching junctions and stationery objects, they are still braking whilst they should be either steering or changing gear. Instead, through the proper application of the System of Car Control, where information encompasses all other phases, the Associate will spot the hazard early, plan for it and seamlessly work through the remainder of the System.
All of the above is why we work with Associates as they initially struggle, but eventually master, the separation of braking and gear changing.
However, there are occasions when this is not practical. Take a left-hand bend at the bottom of a steep hill. No matter how early you start to brake, the moment you engage the clutch (and therefore release engine braking) gravity will force the car to accelerate. Or, consider the situation where a car is following you close behind and you assess that it may be unsafe or confusing to brake too early. In both of these examples, an overlap would be encouraged because it would be planned: information would have been taken in and used to form a driving plan. It would not be rushed, or have the ability to destabilise the vehicle. In short, the System would not have been compromised.
So, the next time you drive, take note of whether you separate out the stages. If you don't, but want to develop this as a skill, my suggestion would be to find a quiet piece of road (usually in an estate) where there is the opportunity to do left and right-hand turns and practice your braking and gear changing. You will find initially that you may be braking too early and in the gear before the turning, and occasionally you may be too late. But, with practice, you will master it. It is certainly something worth working at, and extremely satisfying when you master.