APUs can be used on the ground for cabin comfort (heating or cooling) but they also provide the air for starting the engines. So if it's not required for cabin comfort, and you use ground power during the turn around, you're going to have to turn it on anyway just before pushback because you need it for the start.
The engines can also be started using a ground air cart instead of the APU, but it's a lot easier and takes less time to use the APU.