My problem with analog sticks in FPSes isn’t fine-grained control – most games have zoom, and auto-aim has done a lot to mitigate lack of acuracy. My problem is coarse-grained control – that is, it takes ages in an FPS to turn around at maximum turn speed, whereas a mouse player can rapidly snap around if they are, say, attacked from the side or behind.
I’ve seen some people talk about hacking together some mechanism to try to deal with this using the Steam Controller and Steam Input – I think that it might have been something like a double-tap-to-rapidly-turn, but my impression is that whatever was going on there was more-elaborate than just the combination of an analog stick and a gyro for fine movement.