As previously mentioned, the firing gun analogy is a bad one. When firing a weapon (w/out optics) most teach to concentrate on the front sight post (not the target) until everything else is "blurry" in order to get the best results. So, this would be more in line with the theory of looking at the cue ball last.
You're right, it's a bad analogy. It seems that firing a gun when aligning the front sight with the back sight is all about making sure the gun is perfectly aligned/straight, which would have more to do with looking at the cue stick last, not the cue ball. Looking at the cue ball last makes sure you are hitting the cue ball where you intended to, but it does not make sure that your stick is in a straight line towards the target, and that your stroke is straight.
We don't have two sights on the cue stick to make sure the stick is straight. We judge our alignment as we are getting down on the shot. Once we are down, we are trusting that we are aligned correctly, and we can see our stick in our peripheral vision during practice strokes, which helps us determine if we are stroking straight or not.
Furthermore, when shooting a gun, since the back sight, front sight, and the target are all at eye level at the same time, even though your focus makes the front sight clear, and everything else blurry, you are still keeping your eyes on the target at the same time, you just don't see it as clear.
Imagine if we could trust that the gun was perfectly straight. Then we could just dial in to a specific spot on a target, forget about the gun, and fire. We would hit exactly where we were looking at every time.