So why have I added this guide when the one linked to above already works? Because I almost didn't read it due to the title, as I didn't think I needed to 'set up' a controller that already 'worked' with Steam and was fine in other games like SA1. If you wish you can simply follow that guide to properly enable XBox controller support in the game and have the analog stick work as expected. The approach to to do this is actually already described in the following guide:
So the way to fix this issue is to enable XBox controller support explicitly in Steam.