As someone else said, the driver is in Steam, not SteamOS. Even on a Steam Deck you have to run Steam in desktop mode to have the buttons on the deck work.
Its been a minute since I've been on desktop mode, but aren't they just a trackpad at that point and none of the button/haptic functionality exists outside of moving the mouse and clicking?
That's not true. You get a reduced functionality controller with trackpads that can still be used to start steam back up.