Wouldn’t convenience and economy be as important as prudery in the cane superseding the birch as the everyday instrument of punishment? A cane, if stored properly, will last indefinitely, while a spray-type birch at least will need frequent replacement.
I understand that at Eton, while prefects and ordinary taechers used the cane, the headmaster and his deputy continued to birch bare bottoms well into the 20th century. Clearly removal of clothing before punishment had not become completely taboo in the Thames Valley. Presumably the birch was retained either because it was considered to be more painful or because the ritual associated with it was thought to have a salutary effect.