Take it from people who had to keep a fleet of vehicles operational in the field that firm and smooth applications of the steering brakes is the only way to drive. You might think the Army is awash with cash and drivers were free to beat vehicles into submission without any regard to the damage they did, but that is a totally incorrect conclusion. Good crews looked after their vehicles and those that didn't often found significant displeasure upon meeting the SSM and Maint WO to explain the damage. Incorrect answers often resulted in an escorted quick march on the CO's carpet.
This idea that correct use of the vehicle brakes will cause damage and "that friction creates heat...heat makes drums expand and crack, shoes glaze and oil burn" is frankly hogwash. The brakes on the vehicle (tanks especially) are designed to stop a fully loaded, fully fuelled and fully crewed tank from top speed to zero in a known and very short distance. The tank must be capable of doing this over and over without overheating the brakes. Tactical necessity require the brakes to be especially resilient when they are responsible for both steering and stopping. Adequate cooling in the Sherman differential is provided by about 40 gallons of oil!
The fact that the Sherman (like the M113) use a controlled differential where power from the braked track is applied to un-braked track
requires slipping of the braked shoes over the drum for a smooth and controlled turn to occur. This idea that short applications of full or even partial brake followed by no brake is somehow easier on the brake linings is ridiculous. This jerky action creates stresses in the whole drive line that are just not necessary.
One thing the Sherman tank does not have is pivot steer. If you look under the hood of a M113 you will see a pair of disc brakes on either side of the differential. These are for completely stopping one track and applying all the residual energy to the other, this results in the sharpest possible turn without neutral steer. They even have separate levers for use, they are not for normal steering (similar to a T16). Your suggested driving actions of jerking the tiller bars to facilitate sharp turns appears to be an attempt to emulate that missing low-speed steering option.
The major teaching point for new tank drivers in moving a tracked vehicle at slow speeds in a confined area was to increase the throttle to give the differential some torque to work with. If you are trying to turn and the tank stops or slows too much you need to increase the throttle to complete the turn with the brake applied, not release the brake. A driver must be in control of the tank, not a passenger.
An interesting WW2 film on tank driving:
https://www.youtube.com/watch?v=73Lxw5OQDmc