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.
The desktop mode is running like an app or game, if you hit the steam button the left hand steam menu will open up and let you switch to a different game, go to the home screen, etc.
If you hit the return to game mode shortcut while inside the nested desktop mode, it’ll just exit desktop dropping you back into your home screen or any games you currently have open.