Same problem. In Steam UI the game runs in the background and I hear the game sound. In desktop mode the game runs but inputs are not working correct so I can't proceed further than the first shooting tutorial.
Setting the gamescope command should fix that, but give me a second to test it. I may not have a steam deck yet but I do have the same software stack as I’m running the same build of steam.
Tested, seems to be a bug with gamescope and/or steam’s gamepad mode, might be worth opening a support ticket with valve or a bug report on gamescope’s repo at https://github.com/Plagman/gamescope
Got around to testing, and as my post now says, it seems to be a bug with gamescope or steam’s game ui.
In the meantime, running it from desktop mode with gamescope -w 568 -h 320 -fine -- %command% as your launch parameters for the game [if running from steam] should provide a great experience, just make sure you have the game’s graphics set to fit and 1x as those parameters let gamescope take care of scaling.
If not running from steam, your launch command would be gamescope -w 568 -h 320 -fin -- ./CrossCode