No, game mode is a specific mode made for the Steam Deck and other portables. I believe people have also gotten it to work on the ROG Ally. It replaces the desktop entirely while active, and only Steam games, and games that are added as a shortcut to Steam can be launched. It is not something that is launched from Steam like Big Picture mode, and while it is active, there is no desktop, and no other way to interact with the Deck. Using the desktop normally requires exiting game mode entirely.