With WatchOS switching the side button to bring up control center, I wonder if the new action button will do something similar for iOS control center. The iPhone X-style swipe down from the top right corner still feels clunky and out of reach most of the time, and all the mentioned functions for the action button can be found in control center today. This feels like a good place to do a dynamic island-style UI change too, focused on control center. Maybe a double press or long press? Back when they were still talking about this and the volume buttons as haptic, there were interesting possibilities for swipe gestures too, like the AirPods Pro 2 volume controls. Maybe that could still happen with a real button?
As long as the Action button can be used to launch a Shortcut, you can use Shortcuts to make a menu which slides down and gives you options to do things.