The most likely reason for this is a problem with the JavaScript on the game website used to detect the presence of Flash Player, and not with Flash itself. Since Flash is a built-in component of Chrome, there's nothing separate to install or maintain.
You can confirm by visiting a simplistic Flash-based website like this: http://www.homestarrunner.com/. If you're seeing animation, Flash works. At that point, you'll need to reach out to the game company to let them know that there's a problem. We recommend that content providers use SWFObject2 in general for Flash detection.