You can install GOG games on a normal Steam Deck already. What you want is better integration with their store but Heroic also works great on Deck and can automatically add your GOG games to Steam as well.
I might be crucified for saying this, but I actually have a Claude skill file for that. I have SSH turned on for the Steamdeck, I use Lutris so all the configs are in a database, so I just point Claude to the MD file and the game files and tell it to set it up for me. The main purpose of the skill file is really just to save the SSH, rsync instructions and give it lutris config examples like which version of proton works, etc.)
It’s a little jank to switch to Heroic to install the game but I haven’t had problems once it’s added to Steam (automatically, I might add, just tick the checkbox in Heroic preferences).
Meh I don't want another store app, I want an API that I can use from my store-agnostic open source game launcher / library. They should just officially support efforts like Heroic instead.
Do you really need a Linux client? just download the installers. They announced a Linux client, and I honestly can't see a reason I would want to install 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.