I can't imagine anyone being willing to download the 185 GB Bitcoin database just to play a game. Perhaps 185 GB will seem tiny in the future, but for now i don't think your idea is feasible.
If the gamer will receive some money (satoshi) for participating in distributed mining - why not?
Mining Bitcoin with a CPU is no longer economically feasible, currently earning less than $0.002 per year. Who would pay the gamer to mine, and why?
This is just one of the options to attract users. If the cost of bitcoin rises again in the future, then it will also be profitable. In addition, there is a GPU in the gaming computer besides the CPU. But the most important way to attract users to the game is of course its quality and creativity, the interestingness of the game. This is where I can help developers. I have enough creativity.