Refactoring, fortunately, took less time than I expected; about two hours, I think, and now things just sort of make a lot more sense. :)
I introduced a Laser (I know, slipping back into canon, dangerous!) object and script that controls... the shooty thing. It also tracks the planets, which are their own game objects, and instantiates UI buttons for each one; this way it doesn't matter how many there are or how they change.
The player script also got a change; it was controlling too much UI, and now just handles the arrow keys and delegates control to the Laser when its control panel is invoked. It'll also delegate to whatever ends up controlling dialogue once that's implemented, so this model makes more sense than what I had, which was basically stream-of-consciousness programming.
I'm still fighting with programs trying to get decent video capture and gif conversion... this one isn't great by any means, but it's better than the last one.
