I just noticed another interesting idea that is somewhat controversial. To keep it unbiased try this exercise first.
Slowly touch your nose with your index finger.
Wait for it
What did you find? I found that my brain prefers to use an arc when moving the hand. Next time I tried moving my index finger in a straight line to my nose. I noticed that my hand wavered along the line. I accomplished the task with accuracy but there was a noticeable difference between the two ways of touching my nose. The arc is (to me) much smoother and the preferred method. I can be equally accurate with either method but the arc simply feels better or more natural.
Perhaps the brain prefers this type of integral calculus for making the necessary adjustment during targeting. It can be forced to work a pump action but a pendulum or arc type swinging motion is "preferred" by my brain.
Well I guess you can see the controversy there. Apparently the pendulum swing is preferred by the brain. Oh my, what are all those snooker players going to say
Then too I have an old brain so maybe the pump is better for some
And finally, touching your nose has nothing to do with pool playing, yeah, right.