I had totally forgotten about these settings. By default the controller dead zones are set quite conservatively. One way to notice this is that the steering input is quite strong once it kicks in, making it for instance slightly more difficult to fine-tune your position coming into a turn.

To change the settings, pause the game, go into "Settings", "Controller", then hit "X" to open the advanced controls where the dead zone settings are. I've set mine to 0 & 100 for steering, acceleration, and braking (in other words, I have no dead zones). If you notice problems, for instance that your car's steering is drifting, try increasing the dead zones slightly.