I would guess that I can point with a lot more precision using a mouse than with my eyes. If I'm moderately careful, I can hit a 3×3 pixel area with my mouse, while the saccadic movement of my eyes is covering maybe a 30 px radius around that area.

However, [1] is a nice read that proves me wrong.

[1] https://news.ycombinator.com/item?id=24696315

I would point out that you can only hit that 3x3 pixel area reasonably quickly because you are repeatedly saccading over it and comparing the mouse with it and (very slowly compared to the saccades) adjusting your gross muscular movements to control it. Seems plausible that using just the saccades would be faster by cutting the mouse and your arm out of the loop entirely; you just focus on it, and 'instantly' enough evidence is accumulated about which 3x3 pixel area is the focus of the jittery saccades to pick it.

That might in general be a good idea, but there are situations where it could get awkward - for example, reading in one window and typing in another.