Goo-Gone, is a terrible remover. I thought is was okay, but I found that rubbing my thumb in the right direction works as well. Sometimes I use a little, tincy, tiny, bit of oil ( cooking oil ) from the kitchen. I know thier is a plastic safe oil like substance, but I do not have the name of it. Even I, mentioned oil, it is never a safe way to remove stickies, without damaging, so it is a slow process.
If you could clean up the back and get a print job of some sort done, I am thinking about acylic ( or something stronger, ) with a stamp method,
or iron on tee-shirt method ( which I do not think would go with plastic ),
or even a blow-dryer method ( maybe a really high end 1500 degrees dryer? That is my idea on how to "restore it" otherwise the game art is damaged.
I would subtract the cost of the game, and sell it for the instructions. That is what I do, when I damage something............like those god forsaken cardboard boxes, especially when they have special print on them.