I had issues saving in Yakuza 0 due to having "PROTON_NO_ESYNC=1" set. After removing it, all seems well.
I also run GOG Galaxy via Steam, but I would NOT recommend it. If you use it, I would either separately add the games directly to steam after downloading them OR start them via Galaxy but then close the Galaxy window in the background. Effectively, you can make use of cloud saves like this, but it's an odd hacky way to use Galaxy. This will sadly also disable GOG achievements.
Why? Because GOG Galaxy runs like dirt under proton. It will impact your performance and battery life on the Steam Deck. You can use the Witcher 3 as a simple benchmark. Run it from Galaxy, look at the FPS, then close the Galaxy window in the background. It will run ... better ;) Another way is to look at the load when you run Galaxy (no game) using top. You will see plenty of load generated by it - even on idle.
I am running a MUCH stronger system, so I really don't mind Galaxy that much, but it is sad that they can't either:
1) Release a Linux client, 2) Release the source, 3) Release open APIs ...
It would take the community less than 30 days to land a nice Linux client :/ Yet here we are, years later doing this nonsense :(