Controller works fine now in 3.16-7. Poor steering seems to just be due to a bad game
This failed with an error message about DirectX 11 support if I didn't turn on Wine's D3D support with PROTON_USE_WINED3D=1. The game ran nicely once it finally started running, but the driving was uncontrollable. It was way too sensitive to be able to use they keyboard, and I was stuck turning hard right with my Xbox 360 gamepad; sometimes in reverse only. I tried another Steam Play game to test the controller and it worked fine.
-- I just tried it again, and this time I found myself steering to the left, but forwards. I found one spot on the stick where I could get it to drive straight, but it was so sensitive that it kept wobbling left and right when I didn't change position. There is something wrong with the calibration. I can't try this game on Windows to see if it is the game or Wine/Proton.