Good to know it works with other games, that means it must be controller configs that are messed up.
In Big Picture Mode go to Steam Controller / Controller Configuration / Browse Configs.
Use recommended configuration: Official Configuration for Altitude0: Lower & Faster.
This should reset controller settings if they somehow got messed up. You can also check key layout on Controller Configuration menu. Brakes, turning, boost, everything should be mapped to Steam Controller buttons.
Otherwise I also tested my Steam Controller on Windows 10 - no problems so far.