The reason specialised hardware is used to mine is because it doesn't have to waste processor cycles and memory on a general-purpose operating system, a browser, games and whatever else you might have running in the background on your PC.  Doing all that extra stuff means you aren't hashing as much as you could be.  An ASIC can dedicate itself solely to the task of hashing.  It's more efficient.  

Your idea might result in more decentralisation if it were viable.  But it can't be economically sustainable, because the more efficient specialised hardware would perform better and earn more of the block rewards.  There wouldn't be sufficient incentive for the gamers to waste their resources and not get adequately rewarded.  

Plus, gamers tend to care about their ping.  If you keep lagging and you lose the game as a result of hashing on the side, it's going to negatively affect your gaming experience.