Another method for gyro aim is flick stick, using the right stick to control the direction of your aim (on the left/right axis) and gyro for fine tuning and also up/down axis.

https://youtu.be/CiSS5OsNCNU from the creator explains it (and older gyro controls).

With controller sticks you control the 3D camera only indirectly -- by telling the rotation velocity (in very limited range) and for how long to apply it.

With gyro you have 1:1 proportional camera position input, like with mice.

It's more or less about possibility of developing muscle memory. With (linear) gyro/mice you could sharply snap camera to a point you see on screen without much overshoot. You could turn 180 degrees in split second with eyes closed (actually with gyro people often use flick stick for such big rotations, turning instantly -- but that's besides the point)

With controller stick? Well you could try to time that 180 turn takes 1.5 seconds of holding at full deflection -- good luck developing a feeling for all the speeds inbetween zero and full deflection.

I have found that you have to keep it centered in order to keep it from moving/registering input, so it worked very similarly to an analog stick to me. Am I mistaken?

I guess I just need to try it some more.

If controller isn't being rotated in the moment, then camera should be stationary regardless of controller orientation, yea.

I guess you've used some strange gyro to stick emulation (never heard of such thing, but sounds like it).