Why not an actual Steam Deck? You can install any software you want on it.
You can and it actually works ok. For the sake of the argument you would/could be more 'free to do with your games what you want', because GoG does not do DRM.
You can but having to hook it up to a mouse and keyboard and monitor every time I want to install new games from not-Steam would be a pain.
Out of curiosity, why not use desktop mode without an external mouse and keyboard for that?
You could write a game store interface that works on whatever its normal mode is.
Not saying this is a great solution either, though it beats the trackpad + on screen keyboard. I like to VNC into the deck and do these things.
Set up SSH!