You can open desktop mode inside of gamemode, just like any other game or application. I find it easier to use most desktop apps this way instead of adding them individually to steam.
To do this, download this script. Save it as a script and make the script executable. Finally right click the file and “add to steam”.
Some important notes with this: You can’t exit steam while running the desktop this way, so apps that require closing steam (like steam rom manager) won’t work. There are also a few apps that do weird things, like cryoutilities shows up super tiny this way. But generally most everything works.
My understanding is that game mode and big picture more aren’t the same thing. Game mode is running Wayland and Gamescope, allowing you to get much better performance compared to desktop mode, which is still using Xorg with KDE.
They aren’t, but doing this opens up desktop mode as a nested Wayland desktop giving it the advantages of Wayland, and since steam deck uses game scope to put each game into its own Wayland session it plays nice with steam and shows up as a normal game.