Very necessary.
If you want to increase the height of the pickup, you will need the spring to "release/loosen" and allow the pickup to rise as you are adjusting the pickup screw and allow the pickup to stay at the height you want.
Without the spring, even if you unscrew the side screws all the way up, your pickup is going to stay at the bottom.
Unless if you are using a bigger size screw which also have to be screwed into the side of the pickup then to the wood, then there is a possibility of not using the spring. But then, this method is not a proper way or even a wrong way..