I will just leave this here for anybody else having this issue. So the problem I had is that sometimes the camera rotation and or mouse movement feels like it sticks. I am using a Razer Death Adder v3 mouse on my Mac.
Turns out that the StarterAssets.FirstPersonController has a threshold it checks before actually moving the Cinemachine camera.
I exposed this _threshold as a public variable so it can be tuned in the editor.
// Before:
// private const float _threshold = 0.01f;
// After:
public float _threshold = 0.01f;
Also I found that I prefer to have 0.0 threshold and now this feels like a twitch shooter like good old Quake 3.
Setup: MacBook Air M2, macOS Monterey, Razer Death Adder v3.