The marksman is wrong. That is NOT the Coriolis effect - period. He is flatly wrong. My bet is that there isn't any marksman anywhere that has ever said that. I totally believe that such an argument was put forward on those videos but that doesn't mean that any real marksman said such a thing. It's just completely rediculous on its face. The marksman himself along with his gun and the bullets in it are all traveling along with the planet that they are sitting on when the gun is fired. The momentum is conserved when the bullet if fired. In other words, the path of the bullet is primarily relative to the "stationary" gun. That is to say that the gun is only stationary relative to the path of the bullet, not the whole universe or even to the spin axis of the planet, thus no correction is necessary because the momentum around the earth's axis is conserved after the bullet leaves the barrel.
Now, the spin of the earth does turn the Earth into one gigantic spinning compass and thus Coriolis effect is present any time anything travels in any direction that is not parallel to the spin of the earth. It is possible that on extremely long shots, like those made by the big guns on ships or other artillery could need to correct for the Coriolis effect when extreme precision is needed but its not because the Earth is moving the target underneath the trajectory of the projectile its because of the preservation of angular momentum. If you want to feel this Coriolis effect, go buy a cheap compass and spin it up and then move it around with your hand. The weird resistance you'll feel is the Coriolis effect.
Resting in Him,
Clete