I want to make an official announcement I can reference for Steam Deck compatibility so I can link it from the store page. As of writing, most controllers will work on Native Linux even if they don't on Windows, but there is a big exception for the Steam Deck and Steam Controller.
Steam's peripherals are designed to be accessed through the Steamworks API which I haven't implemented. The intention had been so I can have the same build available on multiple storefronts and Steamworks by design doesn't work properly if Steam isn't running. On Linux this presents an annoying problem because these peripherals don't play by the same rules other controllers do and I don't have a Steam Deck handy to work with.
In the short term, the solution is to just not play the native Linux build on Steam Deck and instead launch the Windows version using Proton. In the long term I plan to eventually just have a Steam-specific version of the game that can have proper Steamworks support.
It's pretty frustrating that developing on Linux-first actually hinders Steam Deck support in this way but that's just how Valve set it up. Controller support was added quite late in development and if I'd known this would be a problem I would have implemented it very differently.