I am pleased to report that Leviathan works great on Steam Deck. (Not a surprise, since it's the same game engine as Meanwhile!)
Left-stick to select a panel, A to advance. Menu button to bring up the menu. Or you can play directly on the touchscreen by tapping panels. That works too.
One possible wrinkle: I have a report(*) that sometimes Steam installs the Windows version of the app on Steam Deck, rather than the Linux version.
If you find that the game dies on launch on Steam Deck, this is probably what's happened. If so, you can work around it by selecting "Properties", "Compatibility", check "Force the Use of Compatibility Tool", and then select "Steam Linux Runtime". This should force Steam to install the Linux version and then Leviathan will run properly.
[* Okay, it was me. This happened to me. I have no idea why. But that's what I did to fix it.]