My basic intuition is that consciousness arises as a byproduct of any sufficiently interconnected (i.e. Complex) network which processes input from the outside world. The problem with AI right now is bottlenecking of both kinds — we don't have computers powerful enough to support the level of complexity of the human brain (100bn neurons with an average of 7k synapses each), and we don't have good enough sensors for capturing the real world (hint: touch is a BIG one).
What sort of game do you have in mind? I definitely think a game in which humans interact with an AI in an artificially-supported information-rich environment (say, the VR of 5-10 years from now) could help provide a suitable environment for "growing" such an AI. Consider the insane amount of Stimulation and calibration human babies require for development! It takes years for the visual system to stablilize, for example.
I think you are underestimating the importance of the motor/action system for the development of consciousness. Many neuroscientists and robotic researchers think that the sensory-motor-environment-sensory loop plays an important role for consciousness. This loop allows you to unsupervised learn all sorts of laws of physics and allows you to think that your body actually is you and not just an extension of your mind and therefore also allows the development of the concept of self awareness. Some researchers might call this sensory-motor-contingencies or embodiment. I think when talking about consciousness there it is very important to take into account this loop and not only the sensory perception and building a neural representation of the world...
The only proof of consciousness case we have (i.e., consciousness arising from Big Bang gases and getting to this conversation) incorporated sense-motor-environment-sense loops. Don't know if it can happen without such interactions, but the one route we know worked, used them.
I definitely should have clarified that point, though, as it's a huge one, I agree!